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Full text of "Nuclear relaxation study of molecular motion in liquid and solid ammonia"

ajCLEAR RELAXATION STUDY OF MOLECULAR- 
MOTION IN LIQUID AND SOLID AMMONIA 



By 
JAMES LYNN CAROLAN 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 

THE UNIVERSITY OF FLORIDA 

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 

DEGREE OF DOCTOR OF PHILOSOPHY 






UNIVERSITY OF FLORIDA 
1969 



ACKNOWLEDGMENTS 

Of all the sections of a dissertation, this is the most difficult 
to write: Difficult because one must for sake of brevity choose a select 
group for recognition from a large number of people who had a profound 
influence en the work. 

To begin, I wish to express appreciation to the members of my doc- 
toral committee: Drs. T, A. Scott, A. A. Sroyles, E, H. Hadlock, C. F. 
Hooper and J. S. Rosenshain, for their many tangible and intangible 
contributions to this work. Special thanks go to the chairman of my 
com.mittee, Dr. T. A. Scott, for suggesting this study and providing a 
very well equipped laboratory in which to perform the work. I wish to 
further thank Dr. Scott- for laboring through the rough draft of this 
manuscript and offering a multitude of helpful suggestions. Special ap- 
preciation is also given Dr. S. S. Ballard, the very able chairman of 
the Department of Physics, vjho first made this work possible by grant- 
ing me a departmental assistantship en an unusually late application. 

Appreciation is also given my many colleagues vjhose contributions 
are manifold. Special thanks go to Mr. P. C. P. Canepa for guidance 
J through the initial confrontation with research electronics, for offering 

numerous very helpful suggestions throughout the course of this work, 
and for use of proton lines that he measured in am:monla at approximately 
1 and 4.2Ko I am also indebted to Dr. James Pl. Srookeraan for many 



XI 



-^^•SiAWBMlIf tM>. Ha^ ft-^ 



interesting and informative conversations, and to Mr. Basil McDowell 
for very able guidance in the art of cryogenic equipment fabrication. 
Among the people who had no influence on what is written in this 
work, but a profound influence on the fact that it exists; I wish to 
J express appreciation to my parents, Mr. and Mrs. James W. Carolan, and 

my wife's parents. Dr. and Mrs. Richard W. Setzer, for continuing en- 
couragement throughout the course of this work. Very special thanks go 
to my wife, Mrs. Joan Setzer Carolan, for encouragement through many 
trying periods, and for careful proof-reading of the manuscript. 

Finally, I wish to thank Mrs. M. Beth Sercombe for her diligent 
proof-reading and highly competent typing of the final manuscript. 



Ill 



TABLE OF CONTENTS 

^ Page 

ACKNOWLEDGMENTS ii 

LIST OF TABLES '"-^ 

LIST OF FIGURES "^iii 

CHAPTER I INTRODUCTION 1 

CHAPTER II THE AMMONIA MOLECULE IN THE SCHEI^E OF Nl-IR 4 

2.1. Molecular Bonding in Ammonia 4 

2.2. Motivation for the Use of Nuclear Magnetic 
Resonance Techniques in the Study of Molec- 
ular Motion 6 

2.3. The Molecular Hamiltonian 8 

2.3.1. The Zeeman Term 8 

2.3.2. Nuclear Dipole-Dipole Interaction 

Term 9 

2.3.3. The Quadrupole Interaction . 12 

2.3.4. Magnetic Interactions of Nuclei with 

Molecular Electrons 14 

2.4. Summary 17 

CHAPTER III INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE AND 

RELAXATION 20 

3.1. Thermal Equilibrium of the Nuclear Spin System ... 20 

3.2. Nuclear Magnetic Resonance Phenomenon 21 

3.3. Continuous Wave vs. Transient N14R 26 

■r-^ 3.3.1. General Remarks 26 

^'4^ ^ 3.3.2. Relationship Between the Lineshape and 

Free Induction Decay 27 

3.3.3. Moments of the Lineshape 34 

3.4. The Effect of Molecular Motion on the Free In- 
duction Decay and Lineshape in Ammonia 35 

3.4.1. The FID and Lineshape in Solid IY6. 37 



iv 



J 



Page 

3.4.2, The FID and Lineshape in Liquid NH- 44 

3.5. The Spin-Lattice Relaxation Time 47 

CHAPTER IV INTERPRETATION OF EXPERIMENTAL RESULTS 51 

4.1. Outline of Experimental Results 51 

4.2. Germane Properties of Liquid and Solid Ammonia ... 52 

4.3. Introduction to Nuclear Relaxation via Molec- 
ular Motion 57 

4.4. Interpretation of Relaxation and Continuous 

Wave Data from Protons in Solid Ammonia 60 

4.4.1. Interpretation of Proton Relaxation 

Data 

4.4.2. Interpretation of Proton Linewidth Data ... 80 

4.4.3. Calculation and Interpretation of the 

Proton Second Moment 84 

4.5. Interpretation of Nuclear Magnetic Relaxation 

Data from 14^ and Protons in Liquid Ammonia 92 

4.5.1. Calculation of Molecular Reorientation 
Correlation Time from Ti Measurements 

on the l^N Nucleus 92 

4.5.2. Interpretation of Proton T-^ and T2 Data 

from Liquid NH , . 100 

CHAPTER V INSTRUMENTATION AND MEASUREMENT PROCEDURE 113 

5.1. Design of Cryostat and Sample Handling Procedure . . 113 

5.1.1. Sample History and Handling Procedure .... 113 

5.1.2. Construction and Performance of the 

Temperature Control Cryostat 116 

5.1.3. Temperature Measurement, Control and 

Stability 119 

5.2. Measurement Procedures Used . .■ 125 

5.2.1. Measurement of T by 90°-t-90° Pulse 

Sequence 125 

5.2.2. The Measurement of the Spin-Spin Relax- 
ation Time 129 

5.2.3. The Measurement of Linewidth and Second 

Moment 135 

.5.3. Electronic Apparatus 136 

5.3.1. Pulsed Spectrometer 136 

Continuous Wave Oscillator 138 

Gated Power Amplifier . - 138 

Pulse Program Circuitry 140 



J 



Page 

Data Recording Device 143 

Magnet 143 

Frequency and Time Measurements 143 

The Coupling Network and Receiver 143 

5.3.2. The Continuous Wave Spectrometer "" 150 

BIBLIOGRAPHY - • ■■ 153 

BIOGRAPHICAL SKETCH 158 



J 



vx 



LIST OF TABLES 



Table Page 

1. The independent contributions to the proton 

second moment for solid NH„ 85 



J 



VI 1 



) 



LIST OF FIGURES - 

Figure Page 

1. Free ammonia molecule 5 

2. The laboratory coordinate system 9 

3. Laboratory and star coordinate systems 23 

4. The rotating components of a linearly polarized 

oscillatory magnetic field 23 

5. 'Ideal' spin system behavior following a 90° pulse .... 29 

6. (a.) Temporal development of M *(t) following a 

90° pulse \ 33 

(b.) Voltage induced in sample coil 33 

(c.) Phase coherent detected sample coil voltage .... 33 

7. (a.) FID (protons of NH at 116 K) 39 

(b.) Solid NH„ lineshape function at three tem- 
peratures 39 

8. - (a.) Liquid ammonia lineshape at 201 K (from ref. 

(19)) 46 

(b.) FID expected from liquid ammonia (protons), 46 

9. Structure of solid NH 54 

EXP '\ 

10. Proton T vs. 10 /(absolute temperature) 61 

EXP "^ 

11. Proton T vs. 10" / (absolute temperature) 62 

EXP 3 

12. Proton T vs. 10 /(absolute temperature) 63 

13. Proton T^ vs. 10 /(absolute temperature) . 64 

14. LN(M recovery) vs. time protons in NH^ 69 

15. LN <R(-t)> vs. tlme/T' 71 



Vlll 



Figure Page 

16. Theoretical proton T reorientation correlation .... 72 

17. Log-_ (hindered rotation correlation time x ) 
vs . temperature 75 

18. Proton linewidth vs. temperature 81 

19. Second moment vs. temperature 86 

20. Log (T^) vs. 10"^/ (temperature) 94 

21. Correlation time t vs. temperature 97 

22. Proton spin-spin relaxation time T„ vs. tem- 
perature 109 

23. Log^Q(T ) vs. 10 /(temperature) Ill 

24. Diagram of the gas handling system 114 

25. The temperature control cryostat 117 

26. Block diagram of the temperature control loop . . .' . . 122 

27. Cryostat heater input power vs. sample chamber 
temperature 123 

28. Sample chamber and temperature bomb 124 

29. Temperature difference vs. sample chamber 
temperature 126 

30. The 90°-T-90° pulse sequence 128 

31. (a.) Sample chamber and magnet coordinate system . . . 132 

(b.) Relative FID amplitude (protons NH„) vs. 

time 132 

32. Block diagram of the pulse spectrometer electronics . . 137 

33. (a.) Configuration used for 90°-t-90° pulse 
y sequence 139 

(b.) Typical data output from 90°-t-90'' experi- 
ment 139 

34. (a.) Configuration used in Carr-Purcell sequence . . . 142 
(b.) Output at points indicated in fig. 34 (a.) .... 142 



Ix 



'\ 



J 



Figure Page 

34. (c.) Typical data output from Carr-Purcell sequence . . 142 

35. Typical parallel tuned sample coil circuit 148 

36. LEL input circuit 148 

37. Arenburg input circuit 148 

38. Block diagram of the continuous wave apparatus 151 



CHAPTER I 



INTRODUCTION 



The problem to be considered in this dissertation is the use of 
experimental nuclear magnetic resonance techniques to arrive at con- 
clusions concerning the character of molecular motion in ammonia 
throughout the temperature range from — 1 to 239 K. More explicitly, 
we obtain numbers characterizing defined behavior of the molecular 
nuclear spin system under certain well-defined conditions; then attempt 
to understand the average microscopic behavior of the molecular system 
using theories reported in the scientific literature. This disserta- 
tion reports work of an experimental nature; thus we avoid reiterating 
a large amount of theoretical material which is handled quite well in 
the extensive literature on the subject. It has been the author's 
experience that the many well-xjritten original publications and re- 
views of the literature by experts in the field are of much greater 
utility than the pseudoreviews one commonly finds in dissertations. 
Some theory is of course necessary for self-consistency, but we will 
adhere to a phenomenological approach with heavy reference to articles 
j^ and texts that the author finds particularly helpful. 

Chap. II is devoted to a discussion of the aimnonia molecule from 
the viewpoint of the radio-frequency spectroscopist , i.e., we will 
introduce the parts of the molecular Hamiltonian relevant to nuclear 
magnetic resonance. We attempt to communicate the motivation one has 



j 



for using nuclear resonance techniques for a study of this nature. 

Nuclear Resonance techniques may be divided into two broad cate- 
gories: Transient and continuous wave NMR. Within the framework of 
continuous wave NMS, the information output of an experiment takes the 
form of a lineshape, or frequency distribution, function; whereas in 
the case of transient NMR one obtains the same information from the 
free induction decay. The intimate inverse relationship in the fre- 
quency and time domains, respectively, between the lineshape function 
and the free induction decay is not often emphasized. We treat this 
relationship in some depth in Chap. Ill after introducing the NMR 
phenomenon via the standard rotating coordinate treatment. A dis- 
cussion of the effect of molecular motion on the lineshape function 
and free induction decay follows, and is illustrated by a comparison 
of theory with experimental results from solid ammonia. We then 
attempt to predict, using previously presented theory, how the ideal 
free induction decay of protons in liquid ammonia should appear. The 
discussion in Chap. Ill is based on the assumption that ideal apparatus 
is available, and this should be kept in mind. Instrumental defects 
can introduce large errors; for. instance, magnetic field inhomogeneity 
is particularly critical in a lineshape study of a liquid, a problem 
to be discussed in some detail in Chap. V. 

In Chap. IV we present and discuss our experimental results. The 
presentation is begun with a general survey of previous experimental 
work on ammonia, the results of which we use to corroborate the model 
of molecular motion adopted in order to explain our experimental re- 
sults. Fortunately, rather specific detailed theories, which fit our 
models of molecular motion nicely, are available from the scientific 



J 



literature. Instead of first introducing the theories and then in a 
separate section applying a theory to a set of experimental results, 
we introduce and give the motivation for use of a particular theory as, 
it is required. It is thought that this procedure is more conducive 
for communication of the process involved in applying the theoretical 
results to our experimental results. 

Chap. V is devoted exclusively to experimental procedure and 
apparatus. We first discuss the apparatus and procedure directly 
connected with handling and temperature control of the sample. The 
actual procedure used in taking and processing data is then explained. 
Finally, we discuss the electronic apparatus used in the experiment. 
For the most part, the basic design and operation of the electronic 
apparatus is fairly straightforward, but there exists one very chal- 
lenging problem which deserves detailed consideration. This is the 
problem of mutually coupling the sample coil to both the power ampli- 
fier and the receiver. Our discussion of this problem is accompanied 
by a number of references to the literature in order that the inter- 
ested reader may further pursue this interesting electronics problem. 



CHAPTER II 

J THE AMMONIA MOLECULE IN THE SCHEME OF NMR 

2.1. Molecular Bonding in Ammonia 

The ammonia molecule consists of three hydrogen atoms bonded to a 
single nitrogen. Fig. 1 illustrates the molecular geometry for the 
isolated molecule (1). Note the C. sjmmietry about the z axis. This 
symmetry is preserved in the lattice structure of solid ammonia. 

Since the molecular structure is the determining factor for 
many of the properties of the compound it seems proper at this point 
to consider the construction of the molecule. Pauling (2) in 1931 
introduced the idea that the direction of a given covalent bond should 
be approximately the direction which yields the maximum overlap of the 
wave functions associated with the bonding electrons of the respective 
atoms. With this in mind we consider the ground state of the nitrogen 
atom. Nitrogen has an atomic number seven and thus three unpaired 
electrons in the 2p state. Recall that the p , p , and p orbitals 
can be pictured as figure eights of revolution, the axes of which lie 
out the X, y, and z axes respectively. Hydrogen has only a single 
V electron in the state Is. Thus with Pauling's theory in mind we would 

expect the three hydrogens to bond to the nitrogen in such a manner that 
each of the 2p electrons forms a spin-up, spin-down pair with one of 
the hydrogen Is electrons. Were this the complete story we would expect 
the three N-H bonds to be mutually perpendicular. This is not the case, 



as is indicated by experiment, and Pauling cites as the cause for the 
discrepancy the partial ionic character of the N-H bond. This leads 
to a net positive charge at each hydrogen and a coulombic repulsion, 
which tends to increase the bond angles. 



>> 



Ca Symmetry Axis 



Nitrogen 




Hydrogen 



Rnh= 1-008 A 
e = 107 



Rhh = '62 A 



Fig. I. FREE AMMONIA MOLECULE 



J 



i 



Another important aspect of this effect, one which has far reach- 
ing consequences, is that the bonded electron wave functions tend to 
concentrate closer to the highly electronegative nitrogen. This re- 
sults in what may be considered a distortion of the 2s orbital forming 
what is known as a lone pair (3) of electrons out the +z axis. This 
high concentration of electron wave function leads to the formation of 
intermolecular bonds of an ionic nature (2) known as hydrogen bonds. 
This type of bonding is responsible for the abnormally high boiling 
point of liquid ammonia and for the bonding of the solid. The hydrogen 
bond energy is estimated by Pauling to be -1.3 kcal/mole. The -93.4 
kcal/mole N-H bond energy (4) gives one some perspective to the magni- 
tude of the hydrogen bond energy. 

2.2. Motivation for the Use of Nuclear Magnetic Resonance 
Techniques in the Study of Molecular Motion 

The objective of this experiment is to study the motion of a typi- 
cal ammonia molecule in the various possible natural physical states 
of the pure substance. This is not a trivial task for a multitude of 
reasons, but primarily because any measurement we may perform neces- 
sarily involves an exchange of energy, thereby altering the normal 
physical state of the system. We must therefore strike the most favor- 
able compromise with Nature: We compromise the ability to study a 
single molecule and look instead at the average motion of an effective- 
ly infinite nxjmber in return for using very low energy techniques. 
Nature has been kind in this respect in that she provides the proper- 
ties necessary to use the smallest possible probe, the nucleus. Many 
nuclei (5) possess spin angular momentum and a magnetic moment and 
those with spin greater than 1/2 also possess an electric quadrupole 



14 

moment. For example the N nucleus has a maximum observable magnetic 

moment of y = .404 in units of the nuclear magneton (y = 5.050 x 10 

erg/gauss), a spin 1=1, and an electric quadrupole moment of 

—26 2 

7.1 X 10 cm . The hydrogen nucleus has a maximum observable mag- 

) netic moment y = 2.7927 y and a spin I = 1/2. It is found that when 

placed in a magnetic field H, the nucleus with magnetic moment y" has 
an energy of orientation 

E = -y-H (2.1) 

Furthermore, only 21+1 states are observed and have energy eigen- 
values 

E = -Yhm|Hi (2.2) 

where 

m= [-1, -d-l),-..., 0, 1] (2.3) 

and Y Is a constant of proportionality, called the gyromagnetic ratio, 
which is determined empirically. The energy of transition for a proton 

in a 5 kG magnetic field from the spin-down to the spin-up state is 

—8 

8.8 X 10 eV and the transition energy in the same field for a bare 

nitrogen nucleus from say a j -1> to a | 0> state is over an order of 
magnitude smaller. The electron also possesses a spin I = 1/2 but the 
transition energy it requires is about 650 times that of the relatively 
strongly coupled proton. Due to the diamagnetic nature of NH_, no 



J 



strong paramagnetic spin or orbital coupling with the applied field 
exists. 

We may thus conclude that the application of a magnetic field to a 
system of ammonia molecules results in a thoroughly trivial perturba- 
tion to the molecular motion and the results obtained by using these 
techniques are quite characteristic of the unperturbed system. 



2.3. The Molecular Hamlltonian 

The Hamlltonian of the isolated ammonia molecule immersed in the 
magnetic field is to be written in a form suitable for the experiment, 
and the individual terms examined in some detail. The general Hamll- 
tonian for an atomic system is written out explicitly in Slichter (6). 
We group all terms in the molecular Hamlltonian which are not relevant 
to the problem at hand into the term Ji . Ironically we are most 

Si 

interested in that part of H describing the motion of the molecule, 
but must abandon it experimentally in order that the temporal develop- 
ment of the part of the wave function in product space governed by 

"H is not disturbed by our measurements. In order for this to be 
a 

true the quantum numbers associated with f\ before we apply the per- 

3. 

turbation to the nuclear spin system must remain good quantum numbers. 
As will be seen subsequently the form of the portion of the molecular 
Hamlltonian that we work with will provide a back door through which 
we are able to extract the desired information. 

The remaining, more relevant terms of the molecular Hamlltonian 
are now written out. Since ultimately we are interested in describing 
molecular motion, we will also briefly consider the effect of transla- 
tion and rigid body rotation on these terms of the Hamlltonian. We 
assume that at the temperatures at which we work the molecules may be 
considered as totally rigid. 

2.3,1. The Zeeman Term 

This term forms the major contribution to the relevant Hamlltonian. 

3 

z ^-, P o 1 'N o N 
J=l 

The first term in Eq, (2.24) constitutes the total proton contribution 



J 



to the molecular energy as defined in Eq. (2.1), and the second term 
is the contribution due to the nitrogen nucleus. H is the applied 
magnetic field. We choose our coordinate system for simplicity such 
that H lies out the +z axis. The observable energy eigenvalues are 
those given in Eq. (2.2). Rotation of the molecule does not affect this 
term, and translation has an effect only if spatial inhomogeneity 
exists in the external field. We assume unless otherwise specified 
that H is spatially and temporally constant. 

2.3.2. Nuclear Dipole-Dipole Interaction Term 



) 



4 4 

" j=i k=i 



dd 



^±:!k _ if?£lik^iiki 






r\ 
3 



(2.5) 



.th 



where y . = y.'fil. is the magnetic moment of the i nucleus r., is the 
3 J J Jk 

vector from nucleus j to nucleus k. This term involves molecular 
coordinates and will prove to be the most important term for obtaining 
the desired information regarding molecular motion. A laboratory 
coordinate system is shotra in Fig. 2 in which the Zeeman Hamiltonian 
can be expressed as a particularly simple diagonal matrix if the proper 
set of basis functions is chosen. In this same coordinate system (7) 
Eq. (2.5) may be expressed in a much more palatable form. 




Fig. 2. The Laboratory Coordinate System 



10 



Eq. (2.5) becomes, 

^dd"2 ^ r - ' 3 [A+B+C+D+E+F] (2.6) 

3=1 k=l r.^ 

2 
where, A = I. I, (1-3 cos e., ) 
jz kz^ jk 

^ B =-i[i;!"i" + i:iJ;](i-3 cos^9.,) = i(i-3 cos^e.,)(i. I, -i.-i) 

^Jk jk jk2 jk^jzkzjk 

C --Ili"!"!, + I. l"^]sin9.,cose. e'^'^jk 
Zjkz jzz jk jk 

D =-|[lTl, + I. irjsine., cose. e^*jk 
2*- J kz jz k' jk jk 

E =--r I.I, sxn 9 ., e J'^ 
4 J k jk 

F=-|i:i:sin2e.,e''*Jk 
4 jTc jk 

and, it = I. +11. 
J jx jy 

I. = I. - il . are the usual raising and lox^^ering operators. 

A particularly illuminating discussion of the individual terms and 

their effect on a system of two spin 1/2 particles is given in Slich- 

ter (6), pp. 45-50. Since this term is so important in what follows it 

is deemed worthwhile to consider in more detail the effect of each 

term of Eq. (2.6) in the case of an isolated pair of identical spin 

1 1/2 particles. ^^^ ^sy be considered a perturbation on the Zeeman 

Hamiltonian. In support of the use of Eq. (2.6) as a perturbation to 

Jx we give some idea of the relative sizes of J~l and J~l, , by calcu- 
z z dd 

J lating the local field one proton in NH„ 'sees' due to a neighboring 

proton on the same molecule . • 

—8 
where r = 1.62 X 10 cm, and 

-24 
y = 2.79 X 5.05 X 10 erg/G. 



J 



11 

Eq. (2.7) gives H = 4,3G whereas H is typically on the order of 
lOkG. 

Term A is of the same form as the classical dipole-dipole inter- 
action. For a given 6., , spin I. sees a local field due to I, which 

either adds to or subtracts from the external field H . Note the 

o 

dependence on 9., . 
jk 

Term B is quite interesting. Written in the form, 

^ = - i^^^k + ^]A)^^ - 3 ^°^^^i^\ (2.8) 

it is evident that it connects one state of zero energy with the 
other; i.e., I. up I down with I up I. down or the inverse. It is 

J K K J 

also referred to as the flip-flop term. The effect of this term which 
also commutes with the Zeeman Hamiltonian is to limit the lifetime of 
a given zero spin state. First order perturbation theory provides the 
mathematical formalism necessary to determine quantitatively its effect 
on the zero spin Zeeman state, but a more qualitative statement of the 
effect of the finite lifetime on the precision of the zero spin state 
may be made using the uncertainty principle. The uncertainty principle 
argument is quite interesting and is developed in detail on page 17 of 
Andrew (8) . 

Terms C and D flip one spin only and thus connect states with both 
spins up or down with zero spin states. 

Terms E and F flip both spins, thus connecting states of total 
spin 1 to states of total spin -1. 

In first order, only terms A and B contribute to splitting of the 
energy levels. Terms C, D, E, and F do produce a second order energy 
shift which is so weak as to be generally ignored if one is concerned 
only with the average dipole contribution to the magnetic field at a 



12 

given nuclear site. These terms are however most important in the 
situation where one is interested in molecular motion because of their 
strong dependence on the polar angles. They have the effect of causing 
transitions among the Zeeman states when the molecule is not station- 
ary. This problem was first considered in the classic Bloembergen, 

; 

Purcell, and Pound paper (9). 

Terms A and B yield valuable information concerning the local 
magnetic field at a nuclear site in a rigid lattice. Van Vleck (10) 
has investigated this problem in detail and derived an expression for 
calculating the mean square local magnetic field at a given nuclear 
site in a rigid lattice xjhen dipole-dipole interactions among all 
nuclei in the solid are allowed. The effect of rapid, isotropic mo- 
lecular motion on terms A and B is to average the angular factor to 

llj 

zero. Theories have been developed to relate the change in the local 
field from the rigid lattice value to molecular motion. 

2.3.3. The Quadrupole Interaction 

To this point we have considered only interactions of a magnetic 
nature, with only brief comment in Sec. 2.2. that the nitrogen nucleus 
possesses an electric quadrupole moment. The quantization of the 
electric quadrupole moment of a nucleus along an electric field gradi- 
ent in space is analogous to the quantization of the nuclear magnetic 
moment along an externally applied magnetic field with one important 
-^ exception: In diamagnetic materials the term in the Hamiltonian due 

to the externally applied magnetic field is the predominant magnetic 
term whereas the molecular electric field gradient usually far exceeds ' 
that which one is able to produce across the sample without breaking 
down the dielectric material. The molecular field gradient at the 



13 



site of a nucleus is generated by the surrounding charges, both elec- 
tronic and nuclear. Excellent derivations of the electric quadrupole 
interaction Hamiltonian are given in Slichter (6) , Abragam (7) , and 
Das and Hahn (11) . The field gradient is expressed as a 3 x 3 tensor ■ 
with non-zero off-diagonal elements when referred to a randomly orient- 
ed coordinate system, but one may diagonalize this tensor with the 
proper choice of molecular coordinate system. By convention the mo- 
lecular coordinate system is chosen so that the principal components 
of the tensor obey the relationship 

IV I > |V I > IV 1. (2.9) 

' zz' — ' yy' — ' xx' 

The field gradient, eq, and asymmetry parameter, ri, are defined by 

Eqs. (2.10) and (2.11). 

eq = V^^ (2.10) 

V - V 
n= % ^^ (2.11) 

zz 

Under these conditions the electric quadrupole Hamiltonian, ^^ , is 

given by 

2 

^Q = fefi^) ^'-"l - ^' ■" h^'-''' -^ ^"')>' <2.12) 

where e is a unit electronic charge, I the spin of the nucleus, Q the 
quadrupole moment of the nucleus, and I and I are respectively the 
raising and lowering operators. 

The ammonia molecule possesses C„ symmetry about the V axis, 
which is identical to the z axis in Fig. 1. It has been experimentally 
determined (12) by X-ray analysis that this rotational symmetry is 
preserved in solid ammonia. This is further supported by quadrupole' 
resonance work (13) in solid ammonia in which a single absorption line 



14 

was found. Thus n = and Eq. (2.12) reduces for the case of the spin 
1 nitrogen nucleus to 

K = 4^ {31^ - I^}. (2.13) 

Q 4 z 

A point of interest in this Hamiltonian is that the I^ operator appears 
J as a squared term leading to degenerate eigenenergles for the m = +1 

states. The factor eq in Eq. (2.13) is dependent on the environment of 
the nucleus whereas the nuclear quadrupole moment, Q, is an intrinsic 
property of the nucleus itself. It is conventional to define the term 
(e qQ/h) as the quadrupole coupling constant expressed in Hz. The 
quadrupole coupling coiistant for the case of nitrogen in ammonia is 
(14) -4.08 MHz in the gaseous state and varies in the solid state from 
-3.16 MHz at 77 K to -3.08 MHz at 193 K (approximately 3 K below the 
normal melting point) . The quadrupole Hamiltonian is thus approxi- 
mately equal in magnitude to the Zeeman Hamiltonian and cannot be 
treated easily as a perturbation as can jLj* 

2.3.4. Magnetic Interactions of Nuclei with Molecular Electrons 

The Zeeman Hamiltonian introduced in Sec. 2.3.1. is rigorously 
accurate only for the case of completely isolated nuclei. We now 
consider the magnetic interactions possible in a diamagnetic molecule 
between the nuclei and the moleciilar electrons. The possible inter- 
actions of this nature are separable into two broad categories: 
a) Interactions that are dependent on the magnitude and in some cases 
the direction of the externally applied magnetic field, and b) Those 
independent of either the magnitude or direction of the applied field. 
Interactions of type a) involve a coupling of the molecular elec- 
trons to the externally applied magnetic field, H^, which has the 



) 



J 



15 

indirect result of adding to or subtracting from H at a given nuclear 
site. Abragam (7)^ p. 175 states that this effect may arise by two 
mechanisms. First the Larmor precession of electronic charges about 
H generates at a nuclear site an effective magnetic field H propor- 
tional to H . Secondly the external magnetic field distorts the elec- 
tronic shells which adds another component H to the field at a nuclear 

P 

site. The contribution of H + H , along H may be function of the 

p d o 

relative orientation of the molecule with H . This effect is usually 

very small and observable only in liquids where the only time average 

of the component of H, + H along H affects the Zeeman Hamiltonian. 

dp o 



Thus we can write 



or more simply 



H -H H -H^ 

H = H + ^-^ + ^-^ (2.14) 

|H I iH I 
' o' ' o' 



H = H^(l - 6^). (2.15) 



The factor 6 is known as the chemical shift and may be either 
positive or negative. The experimental fact that it is different for 
each molecular species is exploited to tremendous advantage by chemists 
in both quantitative and qualitative analysis of unknown substances. 
The chemical shift is generally given referred to some solvent, typi- 
cally tetramethyl silane, and is for protons quite small; e.g., in the 
case of NH_, the chemical shift of the protons in NH relative to the 

J protons in H2O is only 0.43 parts per million ,(15) . The effect of the 

dipolar interaction, x^rhich at H =10 kG causes a shift on the order 
of a few hundred parts per million would obviously totally wash out 
the effect of the chemical shift were it not for the fact that in 

I liquids the dipolar contribution to the magnetic field at a nuclear 



J 



J 



16 

site averages to zero due to rapid molecular reorientation. 

It is possible to obtain information concerning molecular motion 

from this interaction if and only if 6 is a function of the direction 

o 

of H relative to molecular coordinates. In the case of ammonia no 
Information will be obtained from this interaction for reasons that 
will soon become apparent. 

Interactions of type b) prove to be quite interesting in the study 
of ammonia. This interaction was discovered independently by Hahn and 
Maxwell (16) and Gutowsky and McCall (17). The paper by Hahn and Max- 
well is particularly interesting because transient NMR techniques are 
used. It was found experimentally that this interaction was independ- 
ent of sample temperature and applied field and could be represented 
by the Hamiltonian, 

^iss = ^^^1*^2' <2.16) 

where h is Planck's constant, J the indirect spin-spin coupling constant 
expressed in Hz, and I and I. are the spin operators for two nuclei 
on the same molecule. The origin of this interaction, which is known 
as the indirect spin-spin interaction , is dependent on the details of 
the molecular electronic structure. It is sufficient for this work to 
consider this interaction as a communication between two nuclei on the 
same molecule through interaction with the molecular electron cloud, 
i.e., the dipole moment of spin I distorts the electronic structure 
of the molecule, and this distortion produces at the site of nucleus I„' 
a small magnetic field with which it interacts. It is important to 
note that although this is effectively a magnetic dipole-dipole inter- ' 
action between nuclei 1 and 2 it does not depend on the relative ori- 
entation of lab and molecular coordinate frames, and is- thus not 



17 



y 



J 



averaged out by rapid molecular reorientations. There are however two 

circumstances which tend to average the effect of J4, to zero: a) If 

ss 

the lifetime of a given nuclear Zeeman state of either I^ or I is much 
shorter than t = 1/J then the other nucleus 'sees' a time average of 
the magnetic field produced at its site due to the rapidly fluctuating 
nucleus, which averages to zero over the period t = 1/J, or b) If 
either nucleus I^ or I undergoes chemical exchange at a rate, v >> J, 
then the exchanging nucleus will 'see' a fluctuating magnetic field due 
to the fact that the other nucleus will be in different states on 
different molecules. 

The indirect spin-spin coupling constant has been measured for 
ammonia (18,19) to be 

Voton - proton = N-^l H^ 

and J ^ ^ = 143.61 Hz. 

proton - proton ' I 

Obviously this small interaction is totally negligible in the solid 
where the dipole-dipole interaction energy is typically several hun- 
dred times this value. 

2.4. Summary 

Experimentally, we are able to consider the various interactions 
with the respective species of nuclei — nitrogen and hydrogen — as to- 
tally separate experiments because the wide difference in magnitude 
of their respective Zeeman Hamiltonians allows us to adjust our ex- 
perimental apparatus to be sensitive only to small no n- over lapping 
frequency intervals at the center frequency corresponding to a Zeeman 
transition energy. Note that although 



0V a»~w m% o i t. h^e S i.«jrT^.at>fvlo^ lM igr>* 



y 



18 



Voton ^ ^^g 

V . 
nitrogen 

the ratio of the respective Zeeman energies is 

nitrogen 
This is due to the more effective coupling of the proton magnetic mo- 
ment to the magnetic field. When expressed in frequency units, Hz, 
using the Planck relationship 



we have. 



E , ^ = liv u ^ . (2.17) 

photon photon 



V ^ = 13.8 V . 
proton nitrogen 



With this simplification of the problem in mind, we break the experi- 
ment down into two sections a) information obtainable from experiments 
concerned with the nitrogen nucleus and b) information obtainable by 
use of the protons as the probe. 

To summarize, we list the possible interactions that each molecular 
I nuclear species may experience when immersed in an aggregate of identi- 

cal molecules. 

A. The Nitrogen Nucleus 

1.) Intramolecular dipole-dipole interaction with each 

of the three molecular protons. 
2.) Quadrupole interaction with the electric field 
J gradient at the nitrogen site. 

3.) Indirect spin-spin interaction with each intra- 
molecular proton. 
4.) Chemical shift interaction. 
5.) Intermolecular dipole interaction with both protons 



19 



and nitrogen nuclei of neighbor molecules. 
B. The Hydrogen Nucleus. 

1.) Intramolecular dipole-dipole interaction among 
protons. 
j 2.) Intramolecular dipole-dipole interaction with 

. nitrogen. 

\ 

3.) Indirect spin-spin interaction with nitrogen. 

4.) Indirect spin-spin interaction among intra- 
molecular protons. 

5.) Chemical Shift Interaction. 

6.) Intermolecular dipole-dipole interaction with 
both protons and nitrogen nuclei of neighbor 
molecules. 



) 



CHAPTER III 

J INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE 

AND RELAXATION 

3.1. Thermal Equilibrium of the Nuclear Spin System 

As stated in Sec. 2.4., for the case of the nuclear species pre- 
sent in NH^ we may experimentally consider each nuclear system inde- 
pendently. Thus we consider a system of N spins/unit volume (spin I) 
immersed in a magnetic field, H . Let the spins 'weakly interact' so 
as to approach the ambient temperature of the lattice. We attempted 
in Chap. II to give some idea of what 'weakly interacting' means in 
terms of the molecular Hamiltonian. Ultimately the details of the 
interactions must be considered, but for the moment we assume only that 
the time scale of the approach of the spin system to equilibrium is 
short enough that we are able to observe the effects of thermal equi- 
librium vs. non-equilibrium in a reasonable amount of time. 

Appealing to Boltzmann statistics as outlined by Reif (20) pp. 257- 
261 the equilibrium nuclear magnetization' is found to be 

M = N yfi 



^ CTNH 



I 2 krl 2 ^^^^ 2kT 



(3.1) 



V A tremendous simplification of Eq. (3.1) is possible for the usual 

magnitudes of applied fields, H < 20 kG, if one works at temperatures 



-3 

above ~ 10 K, i.e., when 



-^ « 1. (3.2) 



20 



) 



21 



one may write for Eq. (3.1): 

M N 2 2 

^o ^H^= 3kF^ ^^^+1)- (3.3) 

o 

The magnitude of the static nuclear magnetic susceptibility, x , 

o 

is typically 10 to 10~ . Therefore, for a laboratory field of 
10 kG one must detect a superimposed field of ~ 1 microGauss: A non- 
trivial experimental problem using static techniques. Static techni- 
ques are difficult experimentally and not very fruitful when one is 
interested in the study of molecular motion, thus other methods must 
be considered. 

3.2. The Nuclear Magnetic Resonance Phenomenon 

We have already established that the effect to be measured is quite 
small and is likely to require correspondingly sophisticated experi- 
mental techniques. Highly elegant techniques have been devised to 
detect this very small effect: All involving the use of a transducer 
which transforms the nuclear magnetic moment into an electrical sig- 
nal which is amplified in some cases over 10 times, recorded, and 



:em. 



finally interpreted in terms of behavior of the nuclear spin systc 
Thus our problem becomes one of removing a very small electrical sig- 
nal from ubiquitous electrical noise, which we define quite generally 
as any signal which has no correlation with the nuclear signal. This 
problem is taken up in the section on experimental techniques; so for 
now we simply state that it is to our great advantage to work with a 
signal of a frequency on the order of 10 MHz. Hence, we are faced 
with the problem of transforming a static, very small magnetic moment 
through a suitable transducer, which must not affect the nuclear spin 
system, into an electrical signal with a characteristic frequency 



22 

around 10 I'lHz. 

• Nature works with us in our task by providing the spins with angu- 
lar momentum as well as a m.agnetic moment. The equilibrium nuclear 
spin situation is characterized by not only a total static magnetic 

^ moment M but also a total static angular momentum L which is related 

JO o 



to M by 

o •' 



\ = ^\' (3.4) 



These quantities are of course quantiom statistical expectation values 
of quantum mechanical operators and the temporal development of a non- 
equilibrium situation must be handled quantum mechanically. Although 
what follows will appear strictly classical, the motion of the expec- 
tation value of the nuclear magnetization vector has been rigorously 
justified quantum mechanically by I. I. Rabi, et al (21). 

We now consider the motion of the spin system described above 
given that H^ = H^^ and at t = 0, M = M i. The temporal development 
of an angular momentum vector is described by 

dL 



^ = torque = MxH^ (3.5) 



and using Eq. (3.4) we find 



dM - - 

^= yMxH^ . • (3.6) 



If we transform to a coordinate system with a common stationary origin 
but rotating at an angular velocity oJ relative to that above we find, 
) using the relationship on p. 133 of Goldstein (22), 



'dM * 



M*xy(H^ + Y ) • -' ^^'''^ 



IdtJ 

The star indicates a quantity measured relative to the rotating refer- 
ence frame. Now if the condition 



23 



y 




Axis of Sample Coil 



x" 



Fig. 3. THE LABORATORY S STAR COORDINATE SYSTEMS 




rf Amp. 
V{t) =-V.sinu:it 




Sample Coil 



Fig. 4. THE ROTATING COMPONENTS OF A LINEARLY POLARIZED 
OSCILLATORY MAGNETIC FIELD 



24 



US - bi = ~ yH = the Larmor frequency (3.8) 

holds, it is apparent that M is not a function of time. If the two 

coordinate systems were aligned at t = 0, M remains aligned with the 

o 

i axis. Thus we have the situation in which M precesses about H 

o '^ o 

at an angular velocity given by Eq. (3.8). If we have the same situ- 
ation except we enclose the sample in a coil of wire, stationary in the 
lab frame with its axis normal to H , then a voltage is induced in the 
coil due to the rotating nuclear magnetic moment which appears to the 
coil as a sinusoidally varying change in magnetic flux with a funda- 
mental angular frequency given by Eq. (3.8). This coil is the trans- 
ducer, mentioned earlier, from which we obtain the signal voltage. This 
is of course a nonphysical situation as the Boltzmann distribution is 

destroyed when M is not colinear with H and the spin system immedl- 
o o 

ately begins to seek equilibrium. Furthermore, we have assumed that 
the local field is the same constant value at each nuclear site — also 
nonphysical. 

We now consider the behavior of the spin system under the influence 
of a small, laboratory generated perturbation to the Zeeman Hamiltonian. 
Let the perturbation be the application of a voltage, 

V = -V^sincot, (3.9) 

to the coll surrounding the sample. For geometric simplification we 
position the coil as shown in Fig. 3, with its axis along the y axis 
and geom.etrical center at the origin. This voltage produces a reason- 
ably homogeneous linearly polarized magnetic field, 

«! = -^2H^sincjt, (3.10) 

at the sample which can be broken-down into two counter rotating cir- 
cularly polarized field components as shoxra in Fig. 4. That the 



; 



y 



25 

component rotating in a sense opposite to that of the free precession 
of the nuclear spin system has negligible effect on the spin system 

is demonstrated by Abragam (7), p. 21. We consider only the component 

ft 

rotating in such a manner that it lies out the x axis. Our initial 

conditions are such that att=0,M=M^ and both the lab and ro- 

o 

tating frame are superimposed. These initial conditions are experi- 
mentally met by first allowing the spin system to achieve equilibrium 
in the field H = H k, then turning on E . Eq. (3.7) thus becomes 

(f]*=M%[H„+^)t%H^Il. (3.11) 

We define an effective field as measured in the * frame by the equa- 
tion, 

H = (H + -)k + H,i . (3.12) 

e o Y 1 ^ ■^ 

-►* 

The angle that H makes with the z axis is 



r 



= tan 



h 



(H + oj/y)^ 
o 



(3.13) 



Note that 9 = u/l when the resonant condition (Eq. (3.8)) is satisfied. 
We now transform to a ** coordinate frame, which rotates at an angular 
velocity, ^, relative to the * frame. Eq. (3.12) transforms to 



dMi — ** -* Q 

°''' =M xy[H^+-]. (3.14) 



dt 



Analyzing Eq. (3.14) by analogy with the previous case we see that the 

^* 

magnetization vector precesses about H when viewed in the * frame. 

e 

Unless specifically stated, the resonant condition will always 
apply in this work. Thus if we let n = -yH-, i , Eq. (3.14) indicates 
that the magnetization vector, as viewed in the * frame, precesses about 
the -i axis at an angular rate, 

n =-yH-|_. (3.15) 



) 



26 



We now define a 11/2 pulse as the application of voltage given by 
Eq. (3.9), with ca = w , to the coil for the time, t ,„, necessary to 

O TT/Z 

precess the magnetization vector dov7n to the x,y plane. From Eq. (3.15) 
we see that the pulse duration for a 7t/2 pulse is. 



-it/2 - 2^- ^3.16) 



Generalizing, the pulse duration for a t, pulse, where ^ is expressed 
in radian measure, is 

^A = ~l~ • (3.17) 

We have implicitly assumed that H is approximately uniform over 
the sample. For this simple analysis we have assumed the magnitude of 
H^ to be immaterial, but it will soon become apparent that H will 
possess certain lower bounds which depend on the sample studied. 

Thus, the task outlined in the first paragraph of this section is 
complete. To measure the equilibrium magnetization of the spin system: 
We simply allow the spin system to come to equilibrium immersed in an 
applied magnetic field, apply a v/l pulse to the coil thereby rotating 
M down into the x,y plane, then monitor the voltage induced in the 
coil by the precessing nuclear magnetization. From the geometry of 
the coil and characteristics of the electronics, |m [ may be calculated. 

3.3. Continuous Wave vs. Transient NME. 

3.3.1. General Remarks 

The nuclear resonance phenomenon is observed, both in continuous 
wave NMR (CWM-IR) and transient NI-IR (TNilR) , as the response of the 
nuclear spin system to an externally applied perturbation. The differ- 
ence in the two methods arises primarily from differences in magnitude 



; 



) 



27 



of the applied perturbation. In the case of CWNMR one continuously 
applies a very small rotating H field, a field so small that the ther- 
mal equilibrium is not appreciably affected. The absorption of energy 
by the spin system is then measured as a ftmction of H rotation fre- 
quency. This absorption curve when normalized gives one the distribu- 
tion of Larmor frequencies (thus local magnetic fields) over the spin 
ensemble. This may be contrasted with the case of TNMR whereby one 
applies rotating H fields of such a magnitude that the Boltzmann pop- 
ulation of levels may be completely destroyed or even inverted. An 
extensive variety of TNMR techniques has evolved in the preceding two 
decades, but we limit our discussion to the technique characterized by 
the following two distinct experimental steps : 

1) A highly intense perturbation is applied to the spin system, 
viz. a 90° pulse. 

2) The response of the spin system to the perturbation is then 
observed without further perturbation . 

3.3.2. Relationship Between the Lineshape and the Free Induction Decay 

In this discussion we assume that a perfectly uniform external 

field H = H k is available, and that the rotating field H, is per- 
o o 1 

fectly uniform over the sample. Furthermore, we assume that no instru- 
mental effects are Introduced. These effects are to be considered in 
connection with the experimental apparatus and neglecting them for now 
simplifies the discussion. 

Perturbations of the Zeeman Hamiltonian, as outlined in Chap. II, 
produce a distribution g(H) in the z component of the magnetic field 
as seen at the sites of the nuclei. Let g(H) be a normalized 



J 



28 



distribution function centered about H with g(H)dH defined as the 
fraction of nuclei/unit volume which 'see' a magnetic field between 
H and H + dH Gauss. The normalization condition is: 

OS 

/g(H)dH = 1 . (3.18) 

—00 

Specifically we will consider a sample of protons in thermal equi- 
librium immersed in an external field, "b. . Denote the thermal equi- 
librium nuclear magnetization as M = M k. The co5rdlnate systems 

o o ■' 

and initial conditions are the same as considered in Sec. 3.2. A 90° 
pulse of radio-frequency (R-F) voltage at frequency, oj , is applied to 
the coil. Note that only the fraction of spins g(H )dH 'see' an effec- 
txve field H^ = H^i . The fraction of spins g(H +AH)dH 'see' an effec- 



tive field 

-t* Aft A* 

H^ = AHk + H^i , (3.19) 

and will precess about this field for the duration of the pulse. Thus 
the condition that must be met experimentally for a 90° pulse to have 
any significance for the complete spin system is 

|h^| » 6H/2, (3.20) 

where 5H is roughly the width of the distribution function g(H). In 

_ft ^* 

this case H - H^i and essentially all the spins precess as a group 

A* A* 

about H^i down to the +yi axis. This condition is sometimes not 
easily met experimentally when the sample is solid due to the very 
large dipole coupling sometimes present, but in liquids it is usually 
quite easy to satisfy. For this discussion we assume this condition 
is met. 

Fig. 5 (a. ) -depicts the situation as seen in the stationary labora- 
tory frame immediately following initiation of the 90° pulse. Fig. 5(b.) 



29 



(a.) 



J 




\ 



A* k-it," "At 



(c.) 




(e.) 



(d) 



M. 




(f.) 




X' 



T,>Ti<i: 



i.'^^^r, 



(A,* = M.V 



it-i;>T, 



Fig. 5. 'ideal' SPIN SYSTEM BEHAVIOR FOLLOWING A 90° PULSE 



M-w- ■-^"•-^*s«*sn»-^. 



J 



30 



depicts the motion of M as seen in the * frame. Under the influence 

o 

of a 90 pulse, M precesses about the x axis, from the z to the y 
axis. The remaining parts of Fig. 5 depict the re-establishment of 
thermal equilibrium. 

The change in the value of M (t=0) = M i for time, t>0, is brought 
on by two processes, both of which are a manifestation of spin ensem- 
ble migration to thermal equilibrium. We note that statistical ther- 
mal equilibrium requires not only that a Boltzmann population exist 
among the Zeeman states, but also that the spin state fxmction phase 
factors are randomly distributed (random phase hypothesis). A 90° 
pulse has the effect of equalizing the spin state popxilations and cre- 
ating a non-random distribution of spin state function phase factors. 
The distribution of phase factors is such that at completion of the 
pulse the vector sum of the nuclear magnetization/unit volume is 
M(o) = M J . The tv/o processes leading to ultimate destruction of 
M *(t) are: The recreation of a Boltzmann distribution, and the re- 
establishment of the random phase distribution. The effect of both 
processes is implicitly contained in the lineshape function, f(aj). 
The normalized lineshape fimction may be viewed as a precession fre- 
quency distribution function, where f(w)dco is the fraction of spins 
with Larmor precession frequency measured in the * frame to be between 
CO and to + doj. fCu) is symmetric about oj = o. H ^(t) will thus be 
destroyed both by processes that tend to fan the spins out and proc- 
esses that induce Zeeman transitions. At any time, x, M ^(t) can be 
written as 

M *(t) = h I cosuxdN^d), (3.21) 

CO 



y 



31 

where dN (t) = The number of spins/unit volume with precession fre- 
quency in the * frame between co and co + do), and y = magnetic moment/ 
spin. Using the fact that f(to)dco represents the fraction of spins with 
precession frequency between co and co + du, the above sum can be written 
as 

My*(t) = M^G(t), (3.22) 

where 

CO 

G(t) = /f (co)cosojtdco. (3.23) 

.-OS 

Eq. (3.22) contains all the physical information available concerning 
the distribution of Larmor frequencies contained in the spin ensemble. 
It is appropriate at this point to introduce the phenomenological 
equations of Bloch (23) . This set of linear coupled differential equa- 
tions describes the temporal behavior of the nuclear magnetization. 
The theory has been quantum mechanically verified under certain condi- 
tions by Wangsness and Bloch (24) and gives particularly good quantita- 
tive information in the case of liquids. Written in vector notation, 

Bloch' 8 equations are: 

jrj Mi + Mj M-M 

- = Y^ - -^ ^ - -V-°-^^ • (3.24) 

2 1 

Bloch 's equations, for the case of free precession following a 90° 
pulse lead to a Lorentz lineshape function: 

and the half-width at half intensity is 

A = ^. (3.26) 

The constant T^ is called the spin -la ttice relaxation time — the charac- 
teristic time constant describing the motion of a non-equilibrium Zee- 



32 

man level population toward thermal equilibrium with the lattice. T^ 
is the spin-spin relaxation time — the characteristic time constant de- 
scribing loss of transverse magnetization. Note that T„ implicitly 
contains any mechanism which causes loss of transverse magnetization; 
J this includes processes governed by T . Thus the lineshape function 

I as derived from Bloch's equations implicitly contains the effect of the 

finite lifetime of a Zeeman state. We now consider the measurement of 
My*(t). 

Recall that the voltage V(t) induced in the coil, which lies with 
its axis out the y axis of our laboratory frame, is proportional to the 
first time derivative of M (t). Transforming Eq. (3.22) to the sta- 
tionary frame we find 

M (t) = -M^G(t) s±n(i^^t - 90°). (3.27) 

In taking the time derivative of Eq. (3.27) we recognize that Eq. (3.22) 
is a very slowly varying function of time, thus its contribution to 
V(t) is negligible. Eq. (3.22) represents the 'information' we wish 
to extract from the 'carrier', -sinco t. Performing the indicated oper- 
ations we find 

V(t) = aM G(t) 10 sinto t, (3.28) 

. o o o 

i 

where a is a proportionality constant. This voltage might appear on 

an oscilloscope as shown in Fig. 6(b.), depending on the form of G(t). 

In practice, however, V(t) is 'detected' and only the 'information', 
I Eq. (3.22), appears on the oscilloscope screen (see Fig. 6(c.)). We 

assume that phase coherent detection is used so that both positive and 
! negative excursions of M -!;(t) are observable. The trace V, (t) is 

known as the free induction decay , henceforth to be referred to as the 

FID. All the constants of proportionality in Eq. (3.28) are lumped 



33 




ng.S(a). TEMPORAL DEVELOPMENT OF M^t) FOLLOWING A 90° PULSE 



V(t) 




Larmor Frequency rf 

nformation Envelope 

ISO'rf Phase Shift at Crossover 



Fig. 6(b.). VOLTAGE INDUCED IN SAMPLE COIL 




Fig. 6(c.). PHASE COHERENT DETECTED SAMPLE COIL VOLTAGE 



34 



into the constant, a, and we have, 

V^(t) = aG(t) (3.29) 

We examine G(t) in more detail at this point. Using the fact that 
f(a)) is an even function of w and that w = 2itv , we write, 

CO oo 

G(t) = /f(v)cos 27rvt 2TTdv = 2Tr/f (v)EXP(-2iTiv)dv. (3.30) 

—CO — CX) 

Thus G(t) is proportional to the Fourier transform of the Larmor dis- 
tribution function for the spins. Now we rearrange Eq. (3.29) and 
multiply through by EXP(ia)'t). 

EXP(ia)'t) ff (ta)EXP(-iiot)daj = -V rt)EXP(iaj't) (3.31) 

ad 

— 00 

Integrating over all t-space and assuming the order of integration is 
interchangeable, we find. 



/duf(uj) /EXP[-it(aj-oj')]dt = - /V rt)EXP(ito't)dt. (3.32) 
_, a ' d 

•mOO —CO —CO 

Using the well-known integral representation for the 6-function, 

eo 

/EXP|-it(a)-a)')]dt = 2^6(w-aj'), (3.33) 

—00 

and performing the integration over oj in Eq. (3-32), we have, 

00 

f (o)') = V^ /v^(t)EXP(iu)'t)dt . (3.34) 

ZTTa ' d 

—CO 

Thus we have demonstrated that the FID and the distribution function 
are related through the Fourier transform. 

3.3.3. Moments of the Lineshape 

We consider the symmetric normalized lineshape function, f(w), 



35 

previously defined. The n moment of f (oj) is defined by 

00 

M^ = /a)'^f(co)da). (3.35) 

—CO 

Note that all odd moments vanish as a result of the even parity of 
f(a)). The importance of determining the moments of a lineshape is 
twofold: a) Knowledge of the various lineshape moments provides de- 
tailed shape information; b) Van Vleck's (10) classic calculation re- 
veals the equality of the mean square local field at a nuclear site 
and the second moment of the lineshape function. 

By using the following equation, one may calculate the second mo- 
ment at the site of a given nuclear spin I due to dipole-dipole inter- 
actions with neighboring I and S spins. This equation is valid only 
if all spins are in equivalent lattice positions. 

2 3yVi(I+1) (1 - 3cos^9.,)^ 

<AH > = — - — ; y 1^^ 

4 ^ 6 

sum over r . , 

I species 

2 2 
+ hX^(S+l) I , J^ > (3.36) 



sum over r., 
S species 



where r are the respective intranuclear distances. This calculation 
is quite tedious for nuclear configurations of little symmetry. Great 

simplifications in this problem are usually possible however thanks to 

—6 

the rapid r decay of the squared dipole-dipole interaction. One may 

usually truncate the sum after nearest or next nearest neighbors. An- 
other simplification occurs for the case of a powdered sample, for 
which Eq. (3.36) averaged over all angles becomes. 



36 



/ 



<AH^> =|-I(I+1)y^^ I r~^ + -^ys(S+l) I r'^ . (3.37) 



Sinn over sum over 

I species S species 

Experimentally the n moment is usually obtained by CWNMR line- 
j shapes, but it may also in some cases be conveniently obtained by 

transient techniques. The expression from Abragam (7) relating the 
n moment to G(t) (Sq. (3.30)) is: 



M„ = (-i)nfd!!|Itl 



l 



G(0). (3.38) 

t=0 



dt 

This method of determining M is used somewhat in the literature, but 

n ' 

suffers from a quite critical experimental difficulty: G(t) is ob- 
scured at t=0 by the finite recovery time of the apparatus. New exper- 
imental techniques currently in use remove this problem using the tech- 
nique of solid echos (25,26,27,28) to reflect the t=0 shape to an ob- 
servable point in time. 

The important point here is the additional support of the intimate 
relationship between the lineshape and the FID. In the next section, 
we hope to give some indication of what may be expected for the FID 

of protons in solid NH^. 

3 

3.4. The Effect of Molecular Motion on the 
Free Induction Decay and Lineshape in Ammonia 

We begin with the statement of five theorems from Fourier trans- 
form analysis. The proofs to these theorems as well as an excellent 
introduction to Fourier transform analysis are presented in R. Brace- 
well's (29) text. 
Theorem 1 : The Addition Theorem 

If f(v) and g(v) have the Fourier transforms F(t) and G(t), re- 



J 



) 



37 

spectively, then f(v) + g(v) has the Fourier transform F(t) + G(t). 

Theorem 2 : The Shift Theorem 

If f(v) has the Fourier transform F(t), then f(v + v ) has the 

o 

Fourxer transform e " F(t). 
Theorem 3 : The Convolution Theorem 

If f(v) and g(v) have Fourier transforms F(t) and G(t), respective- 
ly, then the convolution of f(v) with g(v); 

00 

f(v)*g(v) = /f(x)g(v - x)dx, (3.39) 

—CO 

has the Fourier transform F(t)G(t). 
Theorem 4 : The Autocorrelation Theorem 

If f(v) has the Fourier transform F(t), then its autocorrelation 
function, 

00 

/f*(u)f(u + v)du, (3.40) 

—OS 

2 

has the Fourier transform |F(t)[ . 

We now define the equivalent width of a function W[f(v)], 

CO 

/f(v)dv 
W[f(v)] = """f^^) , (3.41) 

if and only if f (o) ?^ 0. W corresponds to the width of a rectangle 
having height f(o) and area equal to that under f(v). 
Theorem 5 : E quivalent Width Theorem 

If a function f(v) has a Fourier transform F(t) and f(o) ?^ 7^ F(t), 
then 

W[f(v)] W[F(t)] = 1. (3.42) 

3.4.1. The FID and Lineshape in Solid NH 

We consider only the magnetic resonance of protons in solid NH.. 
The nitrogen magnetic resonance is weak and smeared out so far as to 



y 



38 

be unobservable in our polycrystalline sample immersed in a 9 KG mag- 
netic field. 

One finds that the lineshape obtained from solids at very low tem- 
peratures is, in many cases, satisfactorily represented by a Gaussian 
shape function. The normalized Gaussian curve is given by 

1 - 2 

f (oj) = — - EXP -\ , (3.43) 

A/2tt 2A 

where A is the half-width between points of maximum slope of f(a)). We 
find the associated FID by substitution of Eq. (3.43) into Eq. (3.29), 
i.e. 

V^(t) = B e"*^ ^ /^ (3.44) 

where B is a constant. The value of T^ is taken to be that value of 

time for which V^(t) equals 1/e its value at t=0, i.e. 

/2 
T2 = t=~. (3.45) 

When the temperature of a solid is increased to the point where 
motion is very rapid, the Lorentzian lineshape is sometimes a better 
approximation to the experimental curve. Eq. (3.29) gives the follow- 
ing relationship for the FID when the Lorentzian lineshape function 
(Eq. (3.25)) is used: 

V^(t) = Ae-^/^2^ (3,^g) 

where A is a constant and T^ is defined in Eq. (3.26) as the half-width 
at half-maximum of the normalized Lorentzian lineshape function. 

Fig. 7(a) depicts a comparison of the FID predicted by Eqs. (3.44) 
and (3.46) with the experimental value measured for protons in NH at 
116 K. Fig. 7(b) illustrates f(!:ij) obtained by the method of a^JNl■'IR. 
The theoretical curve was empirically fit as well as possible to the 



39 



— LORENTZIAN 

— GAUSSIAN 
EXPERIMENTAL POINTS 




8 10 12 14 16 18 20 22 24 
TIME (MICROSECONDS) 



Fig. 7(a.), FID (PROTONS OF NH, AT II6K) 



4.2 



I90K 




UJ 

o 



iLl 
> 



cr 
o 

UJ 
N 

< 

q: 
o 

z 



III K 



.•-•• 



■ f - r 



60 50 40 30 20 10 10 20 30 40 50 60 

FREQUENCY (KHz) 



Fig. 7(b.). SOLID NH3 LINESHAPE FUNCTION AT THREE TEMPERATURES 



40 

experimental curve. This comparison is for the sake of illustration 
only. Quantitative results on lineshapes from TNMR when a FID is as 
short as that for protons in NH_, although perfectly feasible, require 
techniques and equipment not used in this laboratory. 

We now consider, qualitatively, the effect of molecular motion on 
a lineshape. Suppose that the FID has been measured at a temperature 
low enough that all motion is essentially of a ground state nature. 
The f (to) measured under these conditions is called the rigid lattice 
distribution function and the width of the distribution between points 
of maximum slope will be referred to as the rigid lattice linewidth . 
The spin-spin relaxation time corresponding to this distribution func- 
tion will be symbolized by T . T sets the time scale in the low 
temperature region: Any perturbation which has a characteristic fre- 
quency less than approximately T„ cannot affect the linewidth because 
the loss of phase coherence among the spins is approximately complete 
before the perturbation can produce any change in the magnetic envi- 
ronment of a given nucleus. To elucidate, we characterize the behavior 
of the precession frequency to(t) of a given spin as seen in the * frame 

by the following parameters: 

1 '''' 
a.) <co> = — / oj(t)dt = average precessional frequency (3.47) 



uj 0) o 
over the time period t . 

b.) T = the average lifetime of a well-defined nuclear 

precessional state. 

c.) CO = the root mean square deviations from <to> 

rms To 

For the low temperature case x >> T. and -o is quite small. 

CO 2r rms ^ 

On the average, precessional states are thus well-defined for many 



41 

times the characteristic time for the complete spin system to lose 

phase coherence. The upper bound on the average lifetime of a pre- 

cessional state for a given nucleus is T , as previously defined. Each 

possible molecular motion can be characterized by the magnitude of the 

fluctuations from <a)> that it is capable of producing at the nuclear 

sites, i.e., we expect different values of co for different types of 

rms ■" 

molecular motion. As the lattice temperature is increased, co is 

rms 

determined almost exclusively by the type of molecular motion activated. 
The average lifetime t decreases with increasing temperature because 
the probability of occurrence of a given mode of molecular motion in- 
creases. In many cases the rate of occurrence of a given mode of motion 
may be represented by a well-defined mathematical expression. The form 
of this expression will be considered in Chap. IV when its effect can 
be explained using data obtained as an example. 

As T decreases (temperature increases) <cj> is not appreciably 
affected until t - "Y. because, as mentioned previously, the important 

time interval is T„ . When the point x == T„ is reached the molecular 

2r cj 2r 

motion becomes effective as a line narrowing mechanism and a linewidth 
reduction occurs. The magnitude of the linewidth reduction depends on 
the details of the motion involved, i.e., the magnitude of the average 
fluctuation it produces. The reasoning here is that very large ran- 
domly fluctuating precession states tend to average <a)> to zero much 
quicker than a smaller random fluctuation. The bulk effect is that the 
average of <aj> over the spin ensemble decreases. The rate of fluc- 
tuatlon due to a given mode of molecular motion is initially an ex- 
tremely rapid function of temperature which tends to level out, ap- 
proaching a limiting value as temperature increases (see Fig. 17). The 



42 

sharpness of the linewidth transition is governad by the temperature 

dependence of the rate of fluctuation when the condition x - T„ 

is satisfied. As t decreases beyond T- , little additional decrease 

w 2r 

in the linewidth occurs due to this process and one is able to define 

another relatively temperature independent T' and t', 

^ 2 w 

Many substances exhibit more than one fairly sharp linewidth transi- 
tion in the solid region between the region of rigid lattice tempera- 
tures and that of the solid-liquid transition. This effect is due to 
the thermal activation of additional modes of molecular reorientation 
which contribute to the averaging process and further narrow the line. 
These additional linewidth transitions occur when the lifetime for the 
prevailing average precessional state becomes approximately equal to 
the prevailing spin-spin relaxation time. 

We have considered in the previous paragraphs the harrowing of the 
normalized lineshape, but have not mentioned the effect of this mole- 
cular motion on the shape of the curve. This is not a trivial problem 
to consider quantitatively, but a very general qualitative statement 
of how the distribution changes may be made. Molecular motion has the 
effect of producing fluctuations in the local magnetic field at nu- 
clear sites. Fluctuations may be placed in one of the two following 
rather vague categories: a.) Fluctuations which have a peak value 
very close to the rigid lattice root mean square local field, and 
b.) Fluctuations which have a peak value much greater than the rigid 
lattice root mean square local field. 

Fluctuations of type a.) are produced by motional processes which 
are not capable of causing much change from the rigid lattice inter- 
nuclear distances. The averaging process in this case is an average 



43 



over angular factors appearing in the dipole-dipole Hamiltonian. One 
observes a line narrowing due to the averaging process previously con- 
sidered, but little or no change of character from an approximate 
Gaussian type shape function is observed. An example of this type of 
narrowing occurs in the case of solid NH when the molecules undergo 
hindered rotation about the C symmetry axis. Note that any intra- 
molecular interaction causes fluctuations of this type when the mole- 
cule undergoes molecular reorientation. 

Fluctuations of type b.) are caused by processes capable of pro- 
ducing large changes from the rigid lattice interproton distances; 
e.g., 1.) molecular collisions in liquids, or 2.) molecular self- 
diffusion in solids via the formation of Frenkel type lattice defects, 
or 3.) molecular reorientation in solids which involves a large re- 
duction in internuclear distances, e.g. isotropic molecular reorienta- 
tion in solid NH„. The large fluctuations in the local magnetic fields 
at nuclear sites produced by these processes tend to increase the prob- 
ability of high energy Zeeman transitions. This leads to a lineshape 
with more intensity in the wings than appears in the Gaussian function. 
The Lorentzian shape function is a much better approximation to the 
physically observed lineshape when the lines are narrowed by fluctua- 
tions of the type b.). 

We need to further emphasize that the previous discussion is of a 
highly qualitative nature and quite incomplete. The intention was 
that of only providing some idea of how molecular motion alters the 
experimentally o bservable lineshapes . 



44 



3.4.2 The FID and Linesh a pe in Liquid NH » 

Pople (30) has considered in detail the effect of N quadrupole 

relaxation on the proton multiplet lineshape in molecules such as NH-- 

14 

If N had a Zeeman lifetime longer than that for the protons, f(a)) 

could be well approximated by a sum of three delta functions . How- 
ever, with fluctuations of the nitrogen nucleus taken into account, 
Pople finds that the protons 'see' an average J which has the effect, 
for moderately slow quadrupole relaxation, of simply broadening the 
6-function lines and producing three Lorentzian lines. The two satel- 
lite lines are broader than the central line, the ratio of half-widths 
at half -maximum being, 

J- : ^ : ~= 3:2:3. (3.48) 

2 2 2 

Consider the definition of the convolution f(v)*g(v), defined in 
Theorem III, with f(v) a Lorentzian Lineshape (Eq. (3.25)) and 
g(v) = 6(v - J). 

f(v)*g(v) = /f(T)6(v-J-T)dT = f(v-J) (3.49) 

The effect of convoluting 5 (v -J) with f (v) is thus simply to shift 
the origin of f(v) from v=0 to v = J. 

We may thus represent Pople 's results, to fairly good approximation, 
when T2J > 1 as 



f (v) = 3 



f(T,,v)*6(v-J) + f(T' v)*5(v+J) + f(T.,v)*5(v=0) 



, (3.50) 



where — : rrr : Tf- satisfies Eq. (3.48), and 
2 2 2 



T_2 dv_ 

— ^ (1/T„)^ + (2ttv)' 



/f(T„,v)dv = /-^ :^ - = i (3.51) 



2' i TT /, ,^ ^2 , ,^_,^2 3 

-2> 



45 



Using the distributive property of the convolution, we find 



f (V) = J 



f(T2.,v)*[5(v-J) + 5(xH-J)] + f(T^,v)*5(v) 



(3.52) 

Substituting into Eq. (3.29) and using Theorems 1 and 2 we obtain 



V^(t) = C 



2EXP(-t/T )cos2TTjt + EXP(-t/T') 



(3.53) 



1 2 1 

where C is a constant. Since — — = T ^jTTj we may write Eq. (3.52) as 

2 2 



V^(t) = C{EXP(-2t/3T2)[2EXP(-t/3T2)cos2TTJt + 1]}. 



(3.54) 



Recall from Eq. (3.48) that (l/T^) is the half-width at half-maxi- 
mum of one of the outside multiplets. J is the splitting from v=0 
expressed in Hz. Fig. 8 (a.) depicts these relationships. 

It is interesting to consider the bracketed part of Eq. (4.54) 
plotted as a function of t for various values of 

1 1 



k = 



T2J 43.6T2' 



(3.55) 



Fig. 8(b.) depicts 



V'(t) = V EXP(-^x43.6kt) 
o J 



2EXP 



43.6kt 



cos(2TTx43.6t) + 1 



, (3.56) 



the numerical values of which were calculated with the Hewlett Packard 
9100A digital computer for k=0.1 and 1.0. 

The purpose of this treatment is to illustrate the form of the FID 
under this particular set of circumstances. Although qualitatively 
correct for k<l, it is completely wrong for k>>l, because the theo- 
retical prediction that the three lines coalese into one narrow Lor- 
entzian, for rapid fluctuations of the nitrogen nucleus, is not taken 
into account. The proper treatment would predict, for increasing k, 
the gradual loss of FID beat pattern and lengthening of T^. 

Transient techniques can in principle resolve J splittings on the 



46 




Fig. 8{a.). LIQUID AMMONIA LINESHAPE AT 201 K (FROM REF.(I9)) 



t \ 



k =0.1 
k = 1.0 




TIME (msec) 



Fig. 8(b.). FID EXPECTED FROM LIQUID AMMONIA (PROTONS) 



47 

order of 

J ^^ . (3.57) 

2 

Thus if ^2^ 1 second, as is the case for many pure molecular liquids, 
TNMR can in principle resolve J-^lHz. In practice one is limited, 
without the use of special high resolution techniques such as spinning 
the sample in the field, by the apparent decrease in T„ caused by mo- 
lecular self -diffusion through an inhomogeneous magnetic field. 

3.5. The Spin-Lattice Relaxation Time 

In Sec. 3.1. an expression was presented for the equilibrium nu- 
clear magnetization. The temperature T appearing in this expression 
was assumed to be that of the 'lattice' in which the spins reside. We 
implicitly assumed that "enough time" had passed, from the time the 

sample was placed in the external field to the time M would be meas- 

o 

ured, for the spin system to achieve a Boltzmann population distribu- 
tion at the temperature of the 'lattice. ' This requires an exchange 
of energy between the spin system and the 'lattice' and implies an 
interaction between the nuclear spins and the 'lattice.* The spin-lattice 
relaxation time T, introduced in Sect. 3.3. provides one with a quanti- 
tative idea of how long one should wait for the spin system to reach 
thermal equilibrium, viz. approximately five times T^ . 

We introduced T in Sect, 3.3. as a parameter in the Bloch equa- 
tions. We now investigate the relation of T^ and the Zeeman transition 
probability and obtain an expression for T^ in terms of the component 
of nuclear magnetization aligned with the applied field. Consider the 
case of N spin 1/2 nuclei per unit volume immersed in a magnetic 
field H = H k. We assume that the heat capacity of the 'lattice,' 



48 



housing the spin system, is much greater than that of the spin system. 
Let W(+-) be the transition probability per unit time from a spin 
aligned to spin antialigned state and W(-+) the inverse. Let N(+) and 
N(-) be the populations of the |+^and j-)> states respectively. At 
equilibrium the condition, 

N(+)W(+-) = N(-)W(-+), (3.58) 

must hold. The spin system obeys Boltzmann statistics; thus the ratio 
of Zeeman level populations is 

(+) ^-E(+)/kT [+|t^H^ + (+|7ftH^)]/kT 



N ^ 

(-) 

e 



N(-) -E(-)/kT ^ ' ^^-^^^ 



where T is the lattice temperature. Thus it follows that 

W(-+) = w(+-)e+^o/'^T . We^«o/kT. ^3^^^^ 

The z component of the nuclear magnetization, M , is thus 

M^ = u[N(+) - N(-)]. (3.61) 

The equation for the temporal development of M is obtained from 

z 

Eq. (3.61), i.e. 

dM 

^ = -2p[N(+)W(+-) - N(-)W(-+)]. (3.62) 



dt 
Using Eq. (3.59), we find 

dM 



-= -2Wy[N(+) - N(-)e^^o''^^]. (3.63) 



dt 

In the case of protons in a 10 kG magnetic field, the exponent in 
Eq. (3.59) has the value, 

f^^ 3.2 X 10"^ 
kT T ' 

Therefore the condition, 



yhH 

-—- « 1, (3.64) 



49 



is satisfied if T > 0.01 K and the exponential function in Eq. (3.63) 
can be approximated very well as the first two terms in its series ex- 



pansion. Using this approximation, Eq. (3.63) may be writt 



en: 



dM 



f^2ii 



But for I = 1/2, Eq. (3.3) yields 

H N 2^2 
= _2_o XJ_ 
kT 4 • 

From Eqs. (3.59) and (3.64) we see that the populations of the two 
energy levels at thermal equilibrium are very nearly equal. Using the 



(3.65) 



(3.66) 



fact that 

N 
N(-) ^ f , 

and Eq. (3.66), we find that Eq. (3.65) reduces to 

dM 

T~ = -2W(M - M ). 
at z o 



(3.67) 



(3.68) 



Eq. (3.68) has the solution, 



M (t) = M 
z o 



i-^Sr^^^^<-'/V 



where we let 



= 2W. 



(3.69 



(3.70) 



Eq. (3.69) although derived by using a number of assumptions and 
approximations, nevertheless provides an excellent working approxima- 
tion for determining T^, even in solids. The most fundamental assump- 
tion used in the derivation of Eq. (3.69) is that which permitted the 
use of statistical techniques, i.e. that between any two spins there 
exists a spin-spin interaction which is weak compared to the Zeeman 
interaction, and on the average is temporally indistinguishable from 
any other possible spin-spin interaction. This assumption is rigor- 
ously satisfied for the magnetic dipole-dipole interaction between 



50 



spins if H^ is much larger than the local magnetic field and all spins 
undergo totally uncorrelated motion. Obviously this is not true for 
the case of more than one identical spin belonging to a rigid molecule 
because the motion of these spins is clearly correlated. A great deal 
of work has been done to determine the effects of correlated spin mo- 
tion on the temporal path of M^ from a non-equilibrium to an equilib- 
rium situation. Including these effects leads to an equation much of 
the same form as Eq. (3.69) but with the single exponential decay re- 
placed by a sum of decaying exponentials with different characteristic 
decay constants and different weighting factors. Eq. (3.69) is, how- 
ever, a very good approximation even in the extreme case of intramo- 
lecular proton relaxation in ammonia via molecular reorientation. 

Another complication to the theory occurs when one considers nuclei 
of spin I > 1/2. This case also leads in general to a solution for 
M^(t) consisting of a weighted sum of exponentials with different time 
constants. This case will not concern us. 



CHAPTER rv 
INTERPRETATION OF EXPERIMENTAL RESULTS 

4.1. Outline of Experimental Results 

We begin with an outline of the actual physical measurements made, 
with brief mention of the method used. A thorough description of ex- 
perimental apparatus and procedure follows in Chap. V. 

I. The Solid State of Ammonia 

A. Measurement of T vs. Temperature for Protons 

1. Region of temperatures studied - 65 to 195 K 

2. Magnetic f ield - 

a.) H = 2.56 kG 
o 

b.) H = 3.87 kG 
o 

c) H = 4.88 kG 
o 

d.) H = 8.22 kG 
o 

3. Method of measurement - 90-T-90 pulse sequence 

B. Measurement of Linewidth of CWNMR Absorption Curve 

1. Region of temperatures studied - --1 to 195 K 

2. Magnetic Field - 1.7 kG 
3. Method of measurement - CWNMR using magnetic 

modulation 

C. Calculation of Second Moment of C1-JNMR Lineshapes 

1. Region of temperatures studied - '-1 to 195 K 

2. Magnetic field - 1.7 kG 

51 • 



) 



52 



3. Method - Calculated from CWNMR lineshapes 
obtained from previous experiment 
II. The Liquid State of Ammonia 

A. Measurement of T^^ vs. Temperature for "'"'^N Zeeman 
Relaxation 

1. Region of Temperatures studied - 195 to 239 K 

2. Magnetic field - H = 8.9 kG 

o 

3. Method of measurement - 90-T-90 pulse sequence 

B. Measurement of T, vs. Temperature for Protons 

1. Region of Temperatures studied - 195 to 239 K 

2. Magnetic field 
a.) H = 0.775 kG 



b.) H = 4.88 kG 
o 

3. Method of Measurement - 90-T-90 pulse sequence 

C. Measurement of T^ vs. Temperature for Protons 

j 

1. Region of Temperatures studied - 195 to 230 K 

, 2. Magnetic field - 4.88 kG 

3. Method of measurement - Carr-Purcell pulse 

sequence 

■^ • 2 . Germane Properties of Liquid and Solid Ammonia 

! All work reported herein was performed at an almost constant pres- 

\ sure over the sample of ~ 20 Ib/in^ absolute. The normal melting and 

; boiling points of NH^ (1) are respectively 195.36 and 239.76 K. The 

anomalously high boiling point for a member of the Group V hydrides 
is attributed to the strong hydrogen bonding characteristics of the 
molecule. 



53 . 



The solid phase crystal structure of m^ and ND^ has been deter- 
mined using single crystal and powder X-ray techniques by Olovsson and 
I . Templeton (0-T) (12) . Measurements were made at temperatures of 77 

and 171 K. The study revealed the crystal space group to be P2 3. 
. The structure is simple cubic with a basis of four molecules, but may 

{ be considered to be a slightly distorted fee lattice with each molecule 

I having 6 nearest neighbors, with which it participates in hydrogen 

bonding, and 6 slightly removed next nearest neighbors. Fig. 9 depicts 
: the normal solid ammonia lattice structure. The view is that looking 

down the trigonal axis toward the origin of a cubic cell. The nitro- 
gen nucleus at the origin does not lie exactly on the Bravis lattice 
point, but 0.0401 1 out the trigonal axis of the cubic cell (out of 
the paper surface). The important point to notice for an understand- 
ing of our work is: Each molecule is hydrogen bonded to six nearest 
, neighbors-all molecules are in equivalent positions. The distance 

between nearest neighbor and next nearest neighbor nitrogen atoms in 
I NH3 was found to be respectively 3.38 and 3.96 £ with the sample at 

171 K. Each molecular free electron pair was interpreted to form 
three hydrogen bonds. A bond is formed with one hydrogen of each of 
three of the nearest neighbors. Each of the three molecular hydrogens 
forms one third of the same type bond with each of the remaining three 
I nearest neighbors. A difference synthesis was also attempted to re- 

J veal the hydrogen positions. The H-N-H angle (107°) was found to be 

I within experimental error of the free molecule value, but the N-H bond 

I distance found (1.13 X) is 12% larger. This bond length is question- 

able because of its unusually large deviation from the free molecule 
value. 



54 



) 




Fig. 9. The structure of solid NH2 , as viewed toward the origin, down 
the Co axis [111]. Hydrogen bonds are indicated by dashed lines. Mol- 
ecules 1,2, 3, 4, 5, and 6 are each hydrogen bonded to molecule 0. The pro- 
ton second moment (Sec. 4.4.1) was calculated for the origin molecule 
proton marked 0; all contributing protons are marked with x. This dia- 
gram is based on work reported in ref. (33). 



■'^'■ ' i* ". ■ 1 '^ . ■ j rf>^ M »» V,rt .'nW^^ll M M T> » t ,4Wl»»i— '^.•»Jst.«\wUj 



55 



The existence of two metastable noncubic forms of solid NH3 has 
been reported in the literature (31,32). The metastable forms occurred 
when NH3 was deposited from the gas phase onto a surface which was kept 
at approximately 77K. Our experiment was performed in such a manner as 
to exclude any possibility of forming a metastable phase, 

A complimentary lattice structure study of polycrystalline ND3 was 
made by Reed and Harris (R-H) (33) using neutron diffraction techniques. 
Neutron diffraction studies give average positions of the nuclei whereas 
X-ray diffraction reveals m.axima in the electron cloud distribution. We 
expect the neutron diffraction m.easurements to be of more use because 
it is the average internuclear distance that is associated with NMR 
measurements. The (R-H) values of N-N distance are within experimental 
error of the (0~T) measurements on ND^, but the N-D bond length of 
1.005 ± 0.023 R is much closer to the free molecule value of 1.008 
* 0.004 g (1,34) than the 1.12 g found by (0-T) . The D-N-D bond angle 
of 110.4° ^ 2.0° is however significantly different from both the (0-T) 
value of 107° and the free molecule value of 107. 4°* 0.2° (34). The devi- 
ation of a hydrogen bonded D from an N-N line was found to be 11.3° 
± 1.7°. 

X-ray powder work has been done on I\fH3 to 4.2K (32) and no crys- 
tal structure change from the high temperature structure is observed. 
Heat capacity measurements from 15K to the vaporization point (35) 
reveal a sharp transition only at the melting point. The heat ca- 
pacity data do display subtle inflections .in the curve at approxi- 
mately 50 and 165 K. These inflections are interpreted to mark the 
onset of different modes of molecular motion, a presumption supported 
by this work. Thermal conductivity of solid NH3 in the temperature 



iit«=£;«iri:*i«a^jF?=* '>'=»^4«-*.*.i«tMp:i.'2bi^A. 



56 

range 23 to 114 K (36) gives no iuformation concerning molecular re- 
orientation, but the thermal expansion data of Manzhelii and Tolkachev 
(37) in the temperature region 24 to 175 K are interpreted taking into 
account hindered rotation of the molecules. The graph of the coef- 
ficient of linear expansion vs. temperature, presented in (37), ex- 
hibits inflection points at 50 and 130 K. These inflection points are 
interpreted by the authors as being due to hindered rotation of the 
molecules. 

Reding and Hornig (38) have measured the infrared absorption spec- 
tra of NH^ and ND^ at 83 K from 300 to 2000 cm""'-. That they find no 
indication of molecular inversion is quite important in the interpreta- 
tion of the results of this work. Hydrogen bonding of the molecules 
in the solid, they assert, raises the barrier to inversion by several 
kilocalories per mole. Torsional vibrational modes which are assigned 
to molecular torsional oscillations about axes normal to the molecular 
symirietry axis occur at 527 cm""'" and 362 cm"""- in NH . A strong absorp- 
tion line was observed at 261 cm" , but not interpreted. 

A quite interesting and important experiment performed by Lehrer 
and O'Konski (14) is the N quadrupole resonance work in solid NH , 
NDH^, ND^H, and ND^. The quadrupole coupling constants were measured 
as a function of temperature from 77 K to the melting point and the 
'quasi-static' value was then calculated to be (3.47 MHz) in both 
m^ and ND^, It is pointed out that torsional vibrations about the 
molecular syimnetry axis, which is also the V^ axis of the field gradient 

c 

tensor, are not effective in reducing the quadrupole coupling constant; 
therefore only torsional vibrations about axes normal to the symmetry 
axis are considered. Bayer theory in conjunction with Reding and 



--w-ITll, . 1-— >»-**> >« r "3VT i .i«^j&-*.».«lH 



^ 



57 

Hornig's (38) values of vibrational frequency gives an excellent fit to 
the experimental data. We interpret these results to mean that little 
reorientation of the NH^ molecule in the solid occurs about axes nor- 
mal to the symmetry axis. Perhaps we should clarify the distinction 
between molecular reorientation and torsional oscillation. The small 
amplitude librations of a molecule within its crystalline potential 
well are referred to as torsional oscillations. Molecular reorienta- 
tion will refer to a physical rotation of the molecule, e.g. a rota- 
tion of the NH^ molecule about its C^ axis through 2^/3 radians— from 
one position of stable equilibrium to another. 

The structure of liquid* NH^ has been determined by X-ray diffrac- 
tion work performed by Kruh and Petz (39). The liquid x^as studied 
under its vapor pressure at 199, 228, and 277 K. Radial distribution 
functions were calculated for all temperatures. They interpret the 
radial distribution function at 199 K to indicate that on the average 
an ammonia molecule has eleven neighbors, seven at a mean distance of 
3.56 S and four at 4.1 S. The liquid structure bears a close resem- 
blance to that of the solid, a manifestation no doubt of the ordering 
effect due to the polar nature of the molecules. One may be led to 
the interesting speculation of a highly ordered liquid state when 
confronted with these results, a speculation not supported by nuclear 
relaxation results. Reorientation in the liquid state is indeed quite 
rapid, 1 yy second, and appears to be of an isotropic nature. 

^•^' Introduction to Nuclear Relax ation vi.a Molecular Motion ■ 

Our objective in this brief introduction is to review for the read- 
er the various contributions to the measured spin-lattice relaxation 



58 



T ,„EXP 
rate, l/T^ . The application of detailed theory to the experimental 

results for ammonia in order to separate the various contributions 
will follow in later sections. We should emphasize that this dis- 
cussion applies both to the solid and the liquid state. The physical 

^ difference between the molecular liquid and solid states is character- 

j Ized by the modes of molecular motion which are thermally activated. 

I It is the determination of what modes of molecular motion are present 

at a given temperature for which we strive. A much more detailed 
account of the problem than that given here may be found in lectures 
by Powles (40) and Bloom (41,42). 

The problem at hand is quite formidable: To separate the various 
contributions to the spin lattice relaxation time, T^^, and interpret 
what mode of molecular motion leads to a specific contribution. The 

1 separation is conveniently effected through the assumption that 

Eq. (3.69) may be written in the form: 



M (t) = M • 
z o 



f M - M(t=0) 

1 - —-^ EXP(-t/Tf^^) 

o 



(4.1) 



where 



EXP ? 1 • ■ (4.2) 

1 ^ ^1 



Proof of the validity of this assumption is not a trivial matter theo- 
retically, but empirical confirmation leads to very good results and 
the assumption is widely used. It is obviously not applicable when 
the recovery of M^(t) is not describable in terms of a single exponen- 



tial. 



For diamagnetic m.aterials, the sum in Eq. (4.2) consists of the 
following four terms : 



59 

. . .intra d , 
1 **'' -'■'-'■1 ~ *^"^ intramolecular dipole-dipole term: This term 

arises from dipole-dipole interactions among the spins in a single mol- 
ecule. It seems quite generally valid to assume that the molecular' 

! 

; bonds are rigid; therefore, magnetic fluctuations which stimulate Zee- 

^ man transitions are caused strictly by fluctuations in the angle e 

I 

(Eq. (2.6)). In solids this term is usually much larger than that due 
to intermolecular dipole-dipole interactions, but in liquids both terms 

are typically of the same order of magnitude. 

„ V ^ .inter d . 

a. J x/i^ - the intermolecular dipole-dipole interaction 

term: The magnetic field fluctuations in this case are produced by 
dipole-dipole interactions (Eq. (2.6)) betx^een spins on different mole- 
cules. If molecular self-diffusion and reorientation are completely 
independent, the effect of this term is governed only by the frequency 
and magnitude of molecular collisions. This term will then contribute 
information concerning molecular self-diffusion. 

SIT 

C.) l/Tj^ - The spin rotation interaction term: The spin rotation 
interaction is the interaction of a nuclear spin magnetic moment with 
the magnetic field produced by a rotating molecule. Fluctuations in 
this magnetic field caused by collision modulated molecular rotational 
states, i.e., changes in the molecular J quantum number, stimulate 
nuclear Zeeman transitions if the fluctuations are rich in the Larmor 
frequency Fourier component. This term is usually important only 
around the critical point of the liquid. For the case of ammonia, 

SIT 

l/T^ has been measured by Smith and Powles (43) and found to be of 
little importance in the normal liquid region. Thus- we are justified 
in not considering the spin-rotation interaction in this work. 

D.) 1/Ti - The electric quadrupole interaction term: If a nuclear 



) 



■ 60 

spin has an electric quadrupole moment and its molecular environment 
is such that a non-vanishing quadrupole coupling constant exists, then 
molecular reorientation causes a reorientation of the field gradient 
tensor (see Sec. 2.3.3.) which produces random field gradient fluctu- 
ations at the nuclear site. Under these circumstances the field gradi- 
ent fluctuations are usioally a much more effective relaxation mecha- 
nism than any other because the quadrupole interaction energy is, in 
most cases, much larger than that of the dipole-dipole interaction. 
Since this relaxation process depends strictly on rigid molecular re- 
orientation, the information one obtains from IjT^ is the same as that 

, ^ . J J- 1 /„intra d 
obtained from 1/T 

The actual separation of the contributions mentioned above must 

await the introduction of suitable physical models with which one may 

mathematically represent the physical situation. We now proceed to 

introduce such models for the most ordered state and then we shall 

continue to the liquid state. 

4.4. Interpretation of Relaxation and Continuous Wave 
Data from Protons in Solid Ammonia 

4.4.1. Interpretation of Proton Relaxation Data 

We first consider the spin-lattice relaxation time data illustrated 
in Figs. 10, 11, 12 and 13 in the form of Log T vs. 10 /T, where T is 
the absolute temperature of the sample. The experimental data points 
are represented by bars which correspond to statistical intervals of 
99% confidence. The method used to process the data is explained in 
the next chapter. 

For the case of solid NH„, Eq. (4.2) may be. written: 



61 



J L 






0> 

o 



T-1 — I — r 



mill I I \ III I I I I I I 1 1 1 1 1 I I I t ,p 



(O 



— G 



T I M i I I — r 



Mil! I — I r 



-TTTTTT-T — T 
O 



t=ss) -1 



-In 



-.^ 



i 


-to 






"5 








u 








^. 








S 


-<M 






o 


•■■ 






» 








x: 








»- 








1 








O 


'^ 


5 






O 


,t- 




•^ 




>. 




■© 




"o" 




"S 






a: 


"— 


-(Ti 




=3 








1- 
< 

s 




-00 




LU 
UJ 




-N 




_l 
O 

< 




-U) 




> 



-in 



o 

I- 
o 

a 



il 



) 



62 



-J ' ' 'MM I — 1 L ! " I I I I I I ■ I I I 1 I I I I I (O 



_in 



.■sf 



c 
o 
a. 



o 



o 



-ro 



_0J 



i 



N 

X 



or 



-<T> 



T I I rmm — i — i r 









— 


-00 








— 








e- 










— ~ 




-N- 




©" 
















-to 




1 1 1 IT 


1 ( r- , 


1 1 1 1 1 1 1 1 1 


-^J- 



UJ 

£r 

I- 
< 

cr 

Q. 

LU 

I- 

LU 

I- 

_J 
O 
0) 
CD 
< 



> 



I 39S) ,^.i 



O 
Q. 



iZ 



63 



J L 



m I I I I t I 



III I I M I I. 



11" I I I I 



) 



©_ 



e 



©r 



N 



e 



) 



O 



c 
o 
a. 



a 
o 



o 
a> 



T— T 



TTTTT- 1 — i r 



TT 



"T-T— 1 — r 



J_U I 1 I I 

o 



(39S)Jl. 



(0 



lO 



ro 



CVJ 



2 < 

-00 



-tr) 



-in 



t-^ 



IT 

r> 
< 

UJ 

OL 

LJ 

ai 

_J 
O 

fS 



CO 

> 



O 

H 
O 

cr 



il 



64 



J I I 



Nil — l__J MM I I \ — I ! M I I I . t f 



N 






-^ 



^ 



c 
o 



O 

o 



o 

V 

I- 

I 
9 



.U3 



in 



.-^ 



fO 



OJ 



-0» 



-co 



-h- 



-co 



-in 



1 — I — r 



rrrm— I — i — i mm— r— r— i — r 






"]]~n I I I — I — r-'^ 
o 



a: 

UJ 
Q. 

■s 

UJ 

t- 

UJ 

1- 

D 
_l 
O 
CO 
CO 

< 



> 



l~- 



o 

q: 
a. 

to 






65 



__L_ = L + 1 + 1 + 1 , (4, 3) 

^EXP ^intra d ^intra d inter d inter d 
1 Ip? Ipn Ipp ipn 

where the letter subscripts refer to proton-proton (pp) and proton- 
nitrogen (pn) interactions. This is a distinct assumption proved by 
experimental results to be valid over most of the temperature region 
studied, i.e., the recovery of M (t) was found to be describable by 

Sq. (4.1) at very nearly all temperatures. Eq. (4.1) was not rigor- 

FXP 
ously valid in the immediate vicinity of the T minimum; this is 

explained later. 

A model describing molecular motion must be chosen by appeal to 
previous experiments. We must consider any process which causes mag- 
netic field fluctuations at the nuclear sites. In order for a process 
to stim.ulate Zeeman transitions, it must have a strong Fourier com- 
ponent at the transition frequency of the nucleus; thus lattice vi- 
brations and molecular torsional oscillations can usually be neglected 
as being of far too high a frequency to induce transitions. Reding and 

Hornig (38) measure molecular torsional frequencies on the order of 

13 

10 Hz whereas the Zaeman transitions for our case occur at fre- 
quencies on the order of 10 Hz. On this basis, torsional oscillations 
and lattice vibrations are neglected; the justification for this ap- 
proximation rests on the final results. Through comparing results 
for other solids (see Abragam (7), pp. 451-58) we are led to considera- 
tion of large scale molecular reorientation. As mentioned in Sec. 4.2. 
each molecule is hydrogen bonded to its six nearest neighbors and the 
bond energy of all bonds is presumably equal. A moment's considera- 
tion of the lattice structure (Fig. 9) will convince the reader that 
the lowest energy m.olecular rotation transition should presumably 






66 



occur about the symnetry axis of the molecule. This presumption is 
well supported by the ^^N quadrupole resonance work of Lehrer and 
O'Konski (14). We thus consider the following model: The proton 
nuclear relaxation is governed exclusively by inter- and intramolecular 
dipole-dipole interactions modulated by the molecule undergoing hin- 
dered rotations about the symmetry axis to the three possible posi- 
tions of stable equilibrium. The reorientation process is assumed to 
be a stationary random process which may be characterized by a correla- 
tion time T^, which is roughly the average time between molecular re- 
orientations. We assume that the process is thermally activated and 
T^ obeys an equation of the form (see Abragam p. 455), 

T^ = T^EXP{E^/RT}, (4.4) 

where t„ is the value of x^ at T = <. , R is the universal gas con- 
stant, and E^ is the activation energy for the process, expressed in 
calories/mole. Eq. (4. 4) is called the Arrhenius equation. For this 
particular case, E^ is- approximately the height of the barrier hinder- 
ing molecular reorientation. 

As a first approximation to Eq. (4.3) we assume all terms on the 
right hand side are negligible but the first. Hilt and Hubbard (H-H) 
(44), using density matrix formalism, have developed the theory for 
nuclear magnetic relaxation of equilateral triangular configurations 
of three spins with I = 1/2 undergoing hindered rotation about the 
symmetry axis. They consider both the cases of a molecule undergoing 
random jumps between three equilibrium positions and a m.olecule under- 
going stochastic rotational diffusion about its symmetry axis. The 
results of the txro cases differ only in the definition of t . We 



67 

assume the first case, where 

T --^ 
c 3v ' (4.5) 

and V Is the probability per unit time for a transition between points 
of stable equilibrium. The calculations involve autocorrelations of 
) each dipole-dipole interaction and cross correlations between differ- 

ent dipole-dipole interactions. Cross correlation effects lead in 
general to the necessity of describing the longitudinal relaxation 
(spin-lattice relaxation) in terms of a sum of decaying exponentials 
rather than a single exponential (45). The cross correlation effects 
are quite small in the case of the molecule undergoing isotropic re- 
orientation (46) , as will be shown later in this work in connection 
with liquid NH3, but may be large for the case of hindered uniaxial 
reorientation (47) . 

With cross correlations taken into account, (H-H) find that M^(t) 
■ after a ./2 pulse is described by a sum of four decaying exponentials 

with different time constants q./T' and different weighting factors 



C., viz. 



where 



4 
M^(t) =MJ1- ICEXP(-q t/T')], (4.^) 

J=l -^ -J 



T' 6, 4^2 

^ = Vo/y« ' ■ (4.7) 

and r^ is the distance between nuclei. Eq. (4.6) is dependent on the 

angle of orientation 3 of the molecular symmetry axis with respect to 

the Zeeman field, and both q.. and C. are tabulated in ref . (44) for 

eleven different values of g. They also numerically average over the 

angles, and provide graphs of the quantity Ln[(M (t) - M )/M ] vs 

2 00 
t/TV. These graphs are convenient for comparison of the theory with 



68 

experimental work on polycrystalline samples. 

Fig. 14 depicts a comparison of this theory with relaxation data 
for protons in solid NH^ at T = 110 K. The data fit the theoretical 
values better if the sum of exponentials in Eq. (4.6) is multiplied by 
a single exponential to take into account, in an ad hoc manner, the 
effects of intermolecular dipole-dipole interactions. One must how- 
ever assume a quite large intermolecular proton-proton interaction to 
obtain a reasonable fit. We are led to believe that observed non- 
linearity may be explainable by the (H-H) theory because the effect 

was strongest, as predicted by the theory, in the region of tempera- 

EXP 
tures close to the T^ minimum. The equipment in this laboratory is 

not suitable for a detailed study of this effect, but we are able 
nonetheless to obtain an analytic expression for t vs . T which is 
thought to be quite accurate. We will also compare our T^^ results 
to the Hilt-Hubbard formulation which neglects cross-correlation 
effects. 

With no cross correlations taken into account (H-H) find M (t) is 

z 

described, following a ^/2 pulse, by the following equation: 



M (t) = M [1 - EXP(-t/T^"^^^ '^)1 
z o Ipp ' ^ ' 



where 



9 Y-S T^ 



^intra d 16 6 
Ipp ^o 



A 9 / 

(1 - cos 6) _j_ (1 + 6cos B + cos 6) 



1 + (u T )' 
o c 



1 + (2aj T ) 
o c 



(4.8) 



(4.9) 



Subsequently we refer to T^ values given by Eq. (4.9) as T^. To com- 
pare the values of T^ given by Eq. (4.9) with our experimental results 

for polycrystalline NH.^, an average of the quantity (M - M (t))/M 

o z o 
must be made over all possible angles B, i.e. 



-<^W'^»*«>«r-yttit[4.iUll» • -««^»U.,it •4>u.' .«N-ai 



69 



; 



TIME (msec) 
2 4 6 8 10 12 14 



16 18 20 




-2.2 

-24- 
-2.6 • 



Theory- (co.i;) = | 



Experiment- -0,= 20.8 MHz 

Temperature- 1 10.2 K 



Fig. 14. LNlMj RECOVERY) vs. TIME— PROTONS IN NH, 



70 



<R(t)> -/ ° -' ' 



\ M 
o 



1 ^ 
= 2 / EXP(-t/T'^)sined3. (4.10) 

o "^ 

This problem required the use of Simpson's numerical integration pro- 
cedure to evaluate the equivalent integral, 

1 
<|l(t)>g = / EXP[-t/T^]dx, (4.11) 

where the substitution, x = cos3, was made. The integral was approxi- 
mated by 10 intervals, i.e., x = 0, 0.2, 0.3,.-., 1.0, and the calcu- 
lation was performed using a Hewlett-Packard 9100A computer. This 
integration procedure was carried out for at least three different 
values of t for each value of o.^t^. For a given value of o^^x^, the 
corresponding T^ was available from the slope of 

Ln<R(t)>g = Ln / EXP[-t/T^]dx (4.12) 

. . o 

vs. t. One may not rigorously define a single T,^ for the M (t) re- 

1 z 

covery expressed by Eq. (4.11). We find however that Eq. (4.12) is a 

quite linear function of t (see Fia 15^ anH a =->r,rr7o t*^ • 1 

0. K. V.OCC rxg. xjj ana a single T governing the 

relaxation is a good approximation. 

Fig. 16 depicts Log tJ, calculated by the procedure above with ' 

r^ =1.651 S, vs. Log t^. The four frequencies used in the experiment 

are represented. One should note that the T^ minima occur at o) t =. 5 

-'• o c 

and that the curves corresponding to different oj^ are indistinguishable 

in the short correlation time limit [(tu t )^ << ll . 

o c ^ * 

We now proceed to obtain an expression for t vs. T . From 

c 1 ^ 

Eq. (4.9) we see that, in the long correlation time limit f((. t )^ » i) 

c ^ o c' -' ' 

T^ is directly proportional to r^, but this is not a. rigorously valid 

expression. Assumption: The actual physical situation is such that, 

in the long correlation time limit, T^^ = kx^ . This assumption is ■ 



71 



x/r 

2 4 6 8 10 12 14 16 18 20 22 24 26 23 30 



\i 


\ 




\ 




\ 


- .2 


%\ IV\_/-vrt/^i 


\\ l".i<J - U.UUI 




A («.t)'= 100 




\\ 




\\ 


- .4 


\ \ 




\ \ 




\ \ 




\ \ 




"^ \ » 


-.6 


\ \ 
\ \ 




\ \ 




\ \ 




\ \ 


.y^ - 8 


\ \ 


•*Os, -o 


\ \ 


2 


\ \ 


Sx- 


\ \ 


z 


\ \ 


_J 


\ \ 


-1.0 


\ \ 




\ \ 




\ \ 




^ \ 




\ \ 




\ \ 


-1.2 


\ \ 




^ \ 




\ \ 




\ \ 




\ \ 


-1.4 


\ \ 
\ \ 




\ \ 




\ \ 


.< 


\ \ 




V \ 


-1.6 


\ \ 




\ \ 




\ \ 




^ \ 




\ \ 


-1 A 


\ ^ 



Fig. 15. LN<(R(t)) vs. TIME-^T' 



72 



I ' ' ' I ' 



n ' I I I i L 



■X,H,I I I ' ' 




o 

in 



. V- 



2 



2 
O 

I- 
< 

LU 
IT 
CC 

o 
o 



< 

h- 
z 

UJ 

o 

Ll) 

(r 



2 
O 

o 
(r 
a. 



b 



< 



I M I I I r 



1 i I I I i — r — r 



(09S) 'i 



11 I I i~r 
o 



cr 
o 

LlI 

X 



CO 



en 

iZ 



73 

quite well justified experimentally in the region of interest as is 

shown by the high linearity of the plot of Log T^^^ vs. 10^/T in the 

low temperature limit. The reader should refer to Figs. 10, 11, 12 

and 13 for support of this statement. Substitution of T^^ = kx into 

1 c 
Eq. (4.4) yields, after some obvious manipulation, 

LnT - LnT + — ^ ~. 4,13) 

10 R ^ 

Using standard linear regression techniques to fit the T^^ data 
points in the linear low temperature region of Figs. 10, 11, 12 and 13 
to Eq. (4.13), one obtains four values of E which should be the same 

3. 

within statistical error. A weighted average over the four values of 
E^ is performed using as weighting factors the inverse of the product 
of a 99% confidence factor with each estimated standard deviation of 
E^. The following value of the activation energy is obtained: 

E^ = 2.30 + 0.02 kcal/mole. (4.14) 

We choose the low temperature limit for this determination for two 
reasons. The foremost reason is that the (H-H) exact theory predicts 
very nearly single T^ recovery in the low temperature limit (see Figs. 
3 and 4 of ref. 44), whereas in the short correlation time limit this 
is not true. Thus we expect our experimentally determined T^"^ to be 
more accurate in the long correlation time limit. Secondly, there is 
additional risk, as the temperature is increased, of thermally acti- 
vating additional modes of molecular motion. In this case the relax- 
ation would be governed by additional correlation times and Eq. (4.13) 
would no longer be valid. 

To determine the constant t°° in Eq. (4.4), we appeal to the best 



'"-=-<-' ift"--- r-^j -^ e™.i4lJr-~.*- itr**-'' 



74 



experimental data obtained (to^ = 2Tr x 20.8 MHz). Before continuing, 

EX? 
we must mentxon that T-i_ was obtained in each case by a linear re- 
gression fit of experimentally obtained values of Ln5.(t) vs. t (R(t) 
as defined by Eq. (4.10)). Furthermore, values of LnR(t) used seldom 
exceeded -2. The reader may easily convince himself that the minimum 
T^ value, obtained from Figs. 4 and 5 of ref. (44) by fitting the best 

straight line through the curves for Ln<R(t)) „ > -2, occurs at 

p 

"o'^c ^ "^' ^^S^' ^ ^^^ ^ represent exact results of the (H-H) theory, 

i.e., when cross-correlation effects are considered. Although our 

T,EXP , 

r^ values are meaningless as an exact recovery constant in the re- 
gion oj^T^ - 1, we nevertheless may rigorously determine the tempera- 
ture for which co^t^ = 1 from the minimum in the curve T^'^ vs. lO^^/T. 

Inspection of Fig. 12 reveals that this minimum occurs at 10"^/T=9. 2+0.1. 

-1 3 
Using T^ = 0)^ at 10 /T and Eq. (4.14) in Eq. (4.4) we obtain a value 

T^ = 1.85 + 0.43 X 10 sec. The confidence limits placed on x 

account for the maximum estimated error in determining both E (Eq. 4.14) 

3. 

3 EXP '^ 

and 10 /T at the minimum of the T^ vs. 10 /t curve. We thus obtain 

the following analytic expression for t vs. T: 

c 

T^ = [1.85 + 0.43 X 10"-^^]EXP[1155/T]. (4.15) 

Fig. 17 depicts Log ^^ vs. T calculated from Eq. (4.15). The in- 
verse half-width betx^een points of inflection obtained from the CWNMR 
data is plotted on the same graph to provide some perspective for the 
reader as to the extremely rapid temperature dependence of Eq. (4.15), 
One obtains an intuitive feeling from Fig. 17 for the sharpness of the 
observed linewidth transitions attributed in the literature to ther- 
mally activated molecular rotation. 



75 



J , 




UJ 

a: 

z> 

< 
ai 

Q. 

S 

UJ 

I- 



(0 

> 






O 

I- 
< 

UJ 

tr 

o 

z 

g 

< 

I- 
O 

a: 

Q 

UJ 

cr 

i 

X 

o" 
o 



((08S) ';, rsOl 






76 



We now return to the additional terms in Eq. (4.3) and attempt to 

EXP 
estimate the effect of each on T . The closest nucleus to a given 

set of protons is the molecular N nucleus, which is positioned a 
distance 1.005 A from each proton. Interactions of this nature con- 
tribute the 1/T, ^ term in Eq. (4.3). To theoretically solve this 
problem exactly for our model would be quite involved and has not been 

done. One would expect a set of coupled equations for the proton M 

z 

recovery such as those in Abragam (7), p. 295. Note that the strength 

2 2 — fi 
of the nitrogen-proton interaction is proportional to y Y r . The 

p n p-n 

second moment expressed in angular frequency has this sam.e form—pro- 
portional to the interaction energy. 1/T is also proportional to the 
interaction energy; therefore, we expect the ratio of the respective 
contributions to the total second moment to give a rough estimate of 
the upper limit of the ratio of respective contributions to the relax- 
ation rate. From a second moment calculation (Sect. 4.4.3.) we find, 

^ ,^intra d 2 intra 

, /intra d " J\ntra = ^-O^^' C^-^^) 

Ipp pp 

The various contributions to the second moment were calculated- using 
the Reed-Harris (33) distances in the corresponding terms of Eq. (3.37)i 
The term 1/Tj_ , corresponding to Eq. (4.16), is over 100 times 
smaller than Eq. (4.16) and is therefore ignored. 

It must be noted, however, that the quadrupole relaxation mecha- 
nism may cause additional fluctuations of the N nucleus which are 
not related to the hindered rotation relaxation mechanism. For ex- 
ample, molecular torsional oscillations about any molecular axis may 
induce Zeeman relaxation of the nitrogen nucleus through the 



) 



11 



quadrupole coupling aiechanlsm, thereby producing fluctuations at the 

proton sites which are totally uncorrelated ^jith hindered rotation. 

14 
We attempted unsuccessfully to find the N pure quadrupole resonance 

in solid NH with the pulse apparatus in order to examine this possi- 
bility. 

Finallv we concern ourselves with the 1/T, term. To this 

Ipp 

writer's knowledge, no detailed theoretical prediction of this term 
has been made for any solid. This would be a quite complex theoreti- 
cal problem and we are content to present only an upper bound esti- 
mate of the effect of this term relative to the intramolecular pro- 
ton-proton interaction. The simple ratio-of-second-moments argument 
used previously is probably a reasonable estimate in this case be- 
cause, at any point during a hindered rotation, the average intermolec- 
ular proton-proton interaction strength remains roughly constant. 
Using this argument, we find 

- ,_inter d „2 inter 
1/T M 

^ 7 - ^7--— = 0.24. (4.17) 

^^^mtra d ^^^2 xntra 

Ipp " pp 
We are able to estimate from Eqs . (4.16) and (4.17) that the maximum 
additional contribution to the total proton relaxation rate from proc- 
esses other than intramolecular proton dipole-dipole interactions is 

approxim.atelv 30%. 

» 

EXP 
With this inform.ation in mind, we re-exaud.ne the Tf data pre- 

j. 

sented in Figs. 10, 11, 12 and 13. The solid black points among 

experimental points were theoretically calculated by first using 

Eq. (4.12), and the approximation outlined, to find T^ for a given 

value of (J T = x, then substitutins; t = (oj ) x into En. (4.15) to 
o c CO . \ / 



78 



3 

find 10 /T. The intramolecular proton-proton distance used is that 

J "^ °^ ^^f- (33) » ^o = 1.651 £. The agreement is quite good in general 

in both the long and short correlation time limits — recall that the 

T,EXP 

T^ poxnts have truly little physical meaning in the region about 

J '^o'^c ~ ^ ^s ^ result of the high non-exponentiality present in the 

recovery. We nevertheless plot them to illustrate the fairly sharp 

minimum from which xj" was determined. The reader should refer to 

j Fig. 12 for illustration of the points discussed since the data taken 

at 20.8 ^ffiz were considerably better than at other frequencies. Note 

EXP 
that the minimum in T^ does not occur at the same point as the 

. . . c 

minimum m T^. The reason for this was explained previously. 

As mentioned previously there is a discrepancy in the literature 

regarding the solid state intramolecular proton-proton separation in 

the ammonia molecule. The X-ray work by Olovsson-Templeton (12) re- 

, veals a value of r^ = 1.817 S whereas the neutron diffraction work by 

Reed-Harris (33) on ND gives the value r = 1.651 1. We wish to 

J o 

offer rather unique support for the (R-H) value through our T^^ 
measurements. One may observe from Fig. 4 of the Hilt-Hubbard (44) 
paper that a single exponential recovery is a very good approximation 

I to the exact theory when ((j^t^)^3< 100. We observe from this figure 

that when (oj^t^) = 100, T^ - 13.2 T', where T^ is the effective 
intramolecular spin-lattice relaxation time and T' is defined by 

J Eq. (4.7). For oj^ = 2^ x 20. 8 MHz we calculate the following values 

for the intramolecular contribution to T : a.) Using the (0-T) 
value for r^ - T^^^ = 109.0 msec, b.) Using the (R-H) value for 



'iRH ~ ^-^'-^ msec. Substituting x = lO/o) into Eq. (4.15), we 



r - T 
O x!\n. ■ ° C """' "o 

calculate lO^/T = 11.2 + 0.2. These calculated values appear as 



79 



horizontal error bars on Fig. 12. Note the large separation of the 
two values. The T^ measurements permit the same high resolution as 
second moment measurements due to the same spatial dependence; i.e., 
the sixth power of the nuclear separation. We have estimated the 
J total contribution to 1/T^ from the sum of Eqs. (4.16) and (4.17) to 

I be roughly 0.30/T^^^^^ ^ Using the value T^^ = 61.4 msec, calcu- 

lated previously, one finds the theoretical prediction for this con- 
tribution to range from 0.04/1^^^ to 0.60/tJ^ with a most probable 
I ^^^"® ^•■'■^''^iPvA- Li'^^wise using the value T^^^ = 109 msec, one finds 

that relaxation mechanisms other than intramolecular proton-proton 
j must contribute from 0.85/1^^^ to 1-86/1^^^ with a most probable value 

of 1.27/Tj^Q^. We see that the (R-H) value of r yields results com- 
patible with both theoretical predictions and our experimental T^^ 
results, whereas the (0-T) values do not. 

The discrepancy in r^ found by the two methods is quite possibly 
not due to experimental uncertainty, but rather a manifestation of 
the different physical quantity measured by the two methods. As men- 
tioned previously, X-ray diffraction measurements give information 
concerning density peaks in the electron cloud, not the nuclear posi- 
tion as does neutron diffraction. Recall that molecular bonding in 
NH^ is such that the molecular electron cloud is heavily concentrated 
about the nitrogen nucleus, leaving the protons relatively bare. In 
^ the solid each proton forms a hydrogen bond with a nearest neighbor 

molecule through mutual electrostatic attraction between the proton 
and the lone electron pair of the neighboring molecules. It seems 
quite feasible that the difference synthesis, performed by Olovssen- 
Templeton, revealed the electron cloud surrounding each proton but 



.; 



) 



80 . 

weighted by the distorted free electron pair distribution of the hydro- 
gen bonded neighbor. This would explain the apparent lengthening of 
the N-H bond. 

4.4.2, Interpretation of Proton Li ne width Data 

We presented in Sec. 3.4.1. a qualitative picture of the phenom- 
enon of linewidth reduction in solids through molecular motion. The 
concept of an average lifetime x of a nuclear precessional state was 

^ CO 

introduced and we stated that motional narrowing begins to occur when 

T - T^. (4.18) 

We also showed that T^ is roughly equal to the inverse of the linewidth 

6u. The condition for the onset of a motional narrowing transition may 

be written, 

T - (oo))""'-. (4.19) 

OJ 

In an absolutely rigid solid, x is governed by spontaneous Zeeman 

OJ 

20 

transitions, viz. x ~' 10 seconds. Molecular motion stimulates 

OJ 

transitions quite effectively, thus reducing x to the average time 

CQ 

between molecular m.otional transitions, x^hich we have labeled x^. 

With these thoughts in mind, consider Figs. 17 and 18. Fig, 18 
depicts the half-width of the CT-Jl^'IR line between points of inflection 
vs. the absolute temperature of the sample. Fig. 17 depicts our 
calculated values of the correlation time for hindered molecular m.o- 
tion about the syminetry axis vs. absolute temperature (heavy line), 
and (6a))~ vs. absolute temperature (light line). The intersection 
of the two curves in Fig. 17 gives one the absolute temperature for 
which the linex-ridth. should begin to undergo a sharp reduction. The 
value obtained in this mianner (''59K) agrees quite favorably with the 
onset of the first linewidth transition as depicted in Fig. 18. We 



81 



J 



J 










^ (£) ^ (NJ O CO 03 

(ZH>i) HiaiM3NI"l NQlOyd 



tr. 

< 

UJ 

a. 

UJ 



> 



9 

2 



Z 

o 

H 
O 

q:: 

Q. 
00 

c: 



82 . 

conclude on this basis that the linewidth transition which is centered 
at 65 K is produced by hindered rotation of the NH molecules about 
their synnnetry axis. A more accurate statement of the origin of this 
linewidth transition will be made in Sec. 4.4.3. 
J One observes from Fig. 18 two additional points of interest dis- 

cussed In Sec. 3.4.1.: a.) The broad, reasonably flat plateau, and 
b.) The occurrence of an additional linewidth transition, beginning 
at T = 170 K, but interrupted before completion by the solid-liquid 

phase transition. The second linewidth transition is of particular 

i 

interest because it marks the activation of a motional process com- 
pletely distinct from symmetry axis hindered rotation. We speculate 
that this new process is either isotropic reorientation of the mole- 
cules, or molecular self-diffusion— note that molecular self-diffusion 
must involve isotropic reorientation. It is not possible to deduce 
precisely what process occurs in this region from the measurements 
1 made; however, one may obtain a very good estimate of the correlation 

time involved from Fig. 17 and Eq. (4.19). By direct analogy with 
the previous case, we consider T - 180 K to mark the temperature at 
which Eq. (4.19) is valid, i.e., a plot of i ^^, the correlation time 
for the second process, vs. absolute temperature will cross the (6(j)"-'- 
curve of Fig. 17 at T - 180 K. This point on the graph corresponds to 

^c2 "^ '^ ^ 10 sec at T = 180 K. (4.20) 

It is also possible to determine the activation energy for this 
second process to an accuracy of approximately 10%. A number of quite 
involved theories have been developed (9,48) to relate the linewidth 
and second moment change at a transition to the correlation frequency 



J 



83 

of the process involved. The theories are difficult to use because 
many very precise measurements are required. Waugh and Fedin (49) 
make suitable approximations to a very simple theory and arrive at 
the following approximate relationship relating the center temperature 
J of a linewidth transition to the activation energy of the process 

involved : 

^a = ^7 T^' (4.21) 

where E^ is expressed in calories/mole and T^ is the absolute tempera- 
ture of the transition midpoint. This relationship agrees within 10% 
of more detailed (but not necessarily more accurate) theories. For 
comparison, Eq. (4.21) gives a value of 2.4 kcal/mole for the activa- 
tion energy of molecular hindered rotation about the symmetry axis; 
a 5% deviation from our previous calculation (Eq. (4.14)). By use of 
Eq. (4.21) we estimate the activation energy of the process causing 
the second transition to be 

E^2 ~ '^•2 + 0.7 kcal/mole. (4.22) 

The question of why the second motional process did not appear in 
the T^ measurements has probably occurred to the reader. Ammonia 
provides a nice case for discussion of the relative sensitivity of the 
two NMR techniques, which we used, to various frequency regions of 
molecular motion. We take this opportunity in answer to the question 
J ■ raised to briefly discuss this point. Comparison of Figs. 16, 17 and 

18 provides one with the key to this question. The linewidth measure- 
ments of Fig. 18 are observed to be extremely sensitive to molecular 
motion onlx in the region where Eq.. (4.19) is approximately satisfied 
(refer to Sec. 3.4.1.) and are quite insensitive to molecular motion 



J 



84 



of other frequency. The linewidth is a characteristic of the solid 
itself, so the frequency region of molecular motion that we may ob- 
serve through C\mifm. techniques is fixed by the characteristics of the 
material. T measurements are much less limited in this respect. We 

note from Fig. 16 that T is a highly sensitive function of to x , and 

-1- o c 

the nuclei couple most effectively to molecular motion when the follow- 
ing relationship holds: 

■"c " ^%^'^- (^-23) 

Eq. (4.23) is to TNMR what Eq. (4.19) is to CI-INMR, but there is one 
very important difference: The value of to is determined uniquely by 
the nuclear gyromagnetic ratio and the externally applied magnetic 
field. Thus the region of x we may study by Tms. is a laboratory 
problem, limited only by how cleverly we design our electronic appa- 
ratus. The literature abounds with clever tricks to study values of 
T^ not usually accessible. A book describing these techniques in 
detail is badly needed, but this is not our purpose and we must trun- 
cate a very interesting subject, only barely introduced. 

4.4.3. Calculation and Interpretation of the Proton Second Moment 

We used the results of a second moment calculation when interpret- 

EXP 
mg the T^ data; the calculation is now presented. The Reed-Harris 

(33) lattice and molecular parameters are used. Since all protons and 

N nuclex are equivalent, we use Eq. (3.37) and consider: Intramo- 
lecular contributions and contributions from as far away as third near- 
est neighbor protons. The geometrical configuration is shown in Fig. 
9, and the various contributions calculated are given in Table I. 



85 



J 



J 



Table I. The Independent Contributions .to the Proton Second 
Moment for Solid NH^ 



Contribution of: 


^^{(P-) 


Intramolecular protons 


35.36 


Molecular N 


2.15 


Nearest neighbor protons (Molecule to which H bonded) 


4.34 


Third neighbor proton (Molecule to which H bonded) 


1.58 


Second neighbor protons (Protons mutually H bonded to 




same N) 


2.64 


Total : 


46.07 G^ 



The fourth nearest neighbor protons are nearly 1 A farther away 

than the third nearest neighbors. We estimate the total neglected 

2 

intermolecular contribution to be less than 2 G . The second moment 

contribution from the intramolecular nitrogen was calculated assuming 
that the predominant contribution to the nitrogen Hamiltonian is from 
the Zeeman effect. This is of course not true in the case of NH ■ the 
nitrogen quadrupole interaction in solid NH„ is roughly equal, at the 
magnetic field available, to the Zeeman interaction. That this may 
lead to error in the calculated second moment is brought out in ref. 
(50). We ignore this effect on the basis that it will probably con- 
tribute nothing additional to our final interpretation of the data. 

The experimentally measured second moment was calculated from 
CWN14R data for a number of different temperatures throughout the solid 
range and was, in each case, corrected for modulation broadening. 
The experimental points are illustrated as solid vertical lines in 
Fig. 19. The points are unfortunately sparse, but we will nevertheless 



86 



' ' ■ ■ ■ 



' ' 



o 

-o 



o 

a. 

1 

>» 

(A 
I 



■g 



o 

■U3 



O 



•CM 3 

< 

UJ 
CL 



O 

■O 



UJ 



UJ 



O 



CO 



ca 



o 



o 



LU 

q: 

I- 
< 

cr 

LlI 
Q. 

LlJ 



V 



O 
CM 



I 

I 

— I 1 — --J- — 1 \ ' < 1 1 1 I 1 I 

ro ig ^ ^ o CO ID 
{,9) 1N3^I0W QN003S 



CM 



o 

CM 



S 



o 



Q 

O 
CJ 
UJ 
CO 

2? 
d> 
iZ 



87 



be able to make a number of important observations. Two additional 
points (circled) were obtained by Gutowsky and Pake (51) and Rabideau 
and Waldstein (52) at temperatures of respectively 90 and 75 K. 

Clearly the most startling contrast with which one is faced is that 
4 between the slowly varying second moment vs. temperature, illustrated 

in Fig. 19, and the abrupt changes in the linewidth vs. temperature, 
illustrated in Fig. 18. Secondly, and just as profound, is the large 
difference in the predicted rigid lattice value of the second moment 
(Table I) and that measured at temperatures as low as the ^He lambda 
point. This large discrepancy is disturbing because our T measure- 
ments extrapolated to this temperature indicate a T on the order of 

25 
10 sec, which is longer even than the spontaneous transition prob- 
ability would indicate. Clearly the T^ measurements predict no hin- 
dered rotation at this temperature contrary to the second moment pre- 
diction that the molecule is rotating at an angular rate greater than 

/• a % 5 —1 

{hbO = 1.8 X 10 sec. . The resolution of this problem requires 
use of concepts not yet fully understood— the concept of quantum me- 
chanical tunneling of the molecule through its hindering potential 
barrier. Considerable effort has been expended at the University of 
Nottingham on problems of this nature. The classic paper, referred to 
many times in the literature, is that of Andrew and Bersohn (53) which 
treats the problem of the effect of rapid classical rotation of a 
J triangular configuration of nuclei on the apparent second moment of 

the configuration. They find that the observable second moment of a 
triangular configuration of spin 1/2 nuclei rotating about the C axis, 
at a rate much faster than the square root of the second moment, ex- 
pressed in frequency units , is one quarter that of the. calculated rigid 



88 



lattice value. Their results seem to work well even in the tempera- 
ture region where classical rotation is impossible, if one assumes 
that molecules in the torsional ground state tunnel through the hin- 
dering barrier at a rapid enough rate. This rather ad hoc approach 
J has recently been placed on firm ground by Allen (54) who treats the 

molecular motion quantum mechanically and finds that indeed for bar- 
riers of energy less than 3 kcal/mole (assumed temperature indepen- 
dent) the classical expression of Andrew and Bersohn should indeed be 
' accurate, even at very low temperatures. Our previously determined 

value of activation energy, E = 2.30+0-02 kcal/mole, places am- 
monia in this category. Allen and Cowking (55) have measured second 
moments and T in some methylbenzenes and obtain results that appear 
much like our Fig. 19, i.e., the long relatively flat plateau from the 
melting point down, with the gentle change of curvature beginning at 
approximately 70 K. One would expect our results to be similar to 
theirs because first, the barriers hindering rotation in the methyl- 
benzenes are very close to that of NH„; and secondly, the inter- and 
intramolecular contributions to the total second moment are roughly 
the same. 

We now proceed to explain in detail our experimental second moment 

values. The Rabideau-Waldstein point at 75 K is within experimental 

2 

uncertainty of our value at 75 K (they quote 16 + 1 G at 75 K.) The 

J 2 

^ Gutowsky-Pake value of 9.7 G at 90 K is too low. This low a value 

\ would require nearly complete averaging of the intermolecular con- 

tribution, a quite unreasonable requirement in view of the type mo- 
lecular motion occurring. We speculate that the state of the art in 
1950 was such that they were not able to detect the reduced intensity 



89 



in the wings, thus substantially reducing their value for M2. Our 
90:1 slgnal-to-noise ratio at 90 K allowed good resolution of the 
wings. The contribution to the total second moment from intramolecu- 
lar proton-proton Interaction is assumed on the basis of ref. (54) 

2 2 
to remain at (35. 36)/ 4 G = 8.85 G . The contribution from the intra- 
molecular nitrogen may be calculated using the reduction factor 

1 2 2 

T-(l - 3cos y) [see Abragam (7), p. 454], where y is the angle between 

the axis of reorientation (C_) and N-H intranuclear vector. Using the 

Reed-Harris molecular parameters, we find y - 71.53° and the reduced 

2 
value of this contribution to be 0.262 G . Thus the total constant 

2 
intra-contribution is 9.1 G . 

We calculated the theoretical second moment contributions using 

the Reed-Harris lattice parameters, measured in solid M)„ at 77 K; 

2 

and we assume these values to be correct for NH„. A value of 6.1 G 

is found experimentally. Thus a very reasonable 29-40% motional 
averaging is required for the experimental and theoretical results to 
agree. Since the condition for maximum motional averaging has long 
since been met, the averaging of the intermolecular contribution 
should remain constant. This being the case, we speculate that the 
observed decrease in M„ vs. temperature, as the temperature is in- 
creased from 77 K, is attributable to linear expansion of the lattice. 
We assume the intramolecular contribution is a constant. The linear 
expansion coefficient was determined in the (0-T) X-ray analysis of 
NH„ to be 11.2 + 0.5 X 10 K . Since the inter-contribution to the 
second moment varies as the sixth pox^er of the internuclear distances, 
we expect that although this is a small effect it may be observable. 
Assuming that all lattice distances expand according to the prescrip- 



90 



tlon, r = r (1 + aAT) , where a is the expansion coefficient and AT is 
the change in temperature, one finds M„ = M / (1 + aAT) . This rela- 
tionship is actually effectively linear in aAT because over the range 
of validity, aAT is so small that terms of order higher than first 
j are negligible. Thus, one has M„ = M^/ (1 + 6aAT) . We used the ex- 

perimental value of M at 77 K as M„ and the (0-T) value for a. The 
thick dotted line in Fig. 19 represents the results of this calculation. 
I The doxraward trend of the experimental points beginning at roughly 

165 K marks the excitation of an additional narrowing process. As 
observed in connection with the linewidth data, this process is a much 
higher energy process than that of hindered C„ rotation. The upward 
trend below 60 K marks the region in which the classical hindered ro- 
tation becomes energetically impossible and the quantum mechanical 
tunneling process becomes important. It appears that thermal expan- 
sion of the lattice provides a plausible explanation of the reduction 
in M„ between 60 and 160 K. 

We must now briefly consider T in the region below 60 K. This 
region was not investigated in the course of this work because the 
author was under the mistaken impression that tunneling phenomena have 
little observable effect on T . Clough (56) pointed out that coher- 
ent tunneling will not effect T , and work by Eades , et al (57,58) 
on methylpentanes indicated experimentally that very long T^ values 
/ were consistent with motionally narrowed lines. The author was un- 

fortunately not ax<rare of the very interesting work of Allen and Cow- 
king (55) which demonstrates multiple T minima at low temperatures. 
Until very recently, no satisfactory explanation of these minima exist- 
ed. This deficiency is apparently resolved in a very recent publica- 



91 



tion by Allen and Clough (59) in which they derive an expression for 
T^ vs. Temperature for the case of tunneling methyl groups. It is not 
unlikely that the same phenomenom will be observed in NH^ and further 

: 3 

' work is planned in the low temperature region of ITO . 

J The contrast between the M and linewldth curves was previously 

mentioned. The high temperature transition is easily explained in 
terms of activation of an additional motional process. This transi- 
tion is in fact reflected by the M^ data. One cannot brush off the 
transition centered at 65 K quite so easily because it does not appear 
to be reflected in the M^ data. One might wonder why a linewidth 
transition occurs at all, in light of our M evidence that C rotation 
occurs at a rate high enough to minimize intramolecular contributions 
at temperatures as low as /^ 1 K. The answer is, no doubt, a manifesta- 
tion of a fundamental difference in the quantum mechanical tunneling 
process vs. that of a thermally activated classical hindered rotation. 
Since the linewidth is. a strong function of this difference, perhaps 
a detailed study of linewidth in the transition region (region where 
the rotational process is predominantly quantum mechanical to that 
where it is predominantly classical) would offer some valuable infor- 
tion concerning the difference in the character of the motion. We 
speculate that the linewidth transition occurs because the motion in- 
volved changes from a predominantly coherent character, at very low 

J 

-^ temperatures, to a predominantly random character at higher tempera- 

tures. 



92 



4.5. Interpretation of Nuclear Magnetic Relaxation Data 
from -^^N and Protons in Liquid Ammonia 

4.5.1. Calculation of Molecular Reorientation Correlation Time 
from T -. Measurements on the I'^N Nucleus 

The liquid state of any substance is characterized by an increase 
J in the number of degrees of freedom and relative magnitude of molec- 

xilar motion. Although X-ray data seem to point to a 'quasi-lattice' 
type liquid structure for ammonia, we present evidence that indicates 
a typical isotropic molecular reorientational time scale of approxi- 
mately 1 picosecond. This is not to imply that a 'quasi-lattice' 
description, in which one assumes the liquid state to be such that a 
molecule is held relatively securely for a short time before 'jumping' 
rapidly to a vacent lattice position, is invalid for the case of NH„. 
Quite to the contrary, a theory of this type should be well suited to 
describe a strongly bonded polar liquid such as ammonia. Our results 
do show, hoxjever, that the time scale for short range order in liquid 
ammonia has an upper bound of roughly 1 picosecond. 

Our objective at this point is to calculate values for the molec- 
ular reorientation correlation time. The fact that, in the NH mole- 

14 
cule, there exists a relatively large N quadrupole coupling constant 

14 
(Sec. 2.3.3.) allows one to neglect all contributions to the N 

Zeeman relaxation time (Eq. (4.2)) but the quadrupole term. This 

technique for separating the molecular rotational correlation times 

gives very good results and is discussed in detail by Moniz and 

Gutowsky (60) and Abragam (7) . One assumes that the nucleus reorients 

at a rate much faster than the inverse of the molecular quadrupole 



93 



coupling constant and thereby averages the quadrupole contribution to 
the molecular energy to zero. This averaging process occurs because 
the molecular reorientation is so rapid on the time scale set by the 
quadrupole coupling constant that the nucleus cannot follow the field 
gradient. When an external magnetic field H is applied, one has the 
usual Zeeman spectrum with very rapid transitions occurring among the 
states due to the quadrupole coupling. 

The Larmor precession frequency in our magnetic field (H = 8.9 kG) 
is oj = 2Tr X 2.74 MHz. Clearly from our work in solid NH^, our meas- 
urements are made in the short correlation time limit, i.e., tu x << 1. 

o c 

14 

For the N nucleus (I = 1) , perturbed by an isotropic rapidly re- 
orienting sjTnmetric field gradient tensor, we find the following equa- 



tion for 1/T^ in Abragam (7), p. 314: 



ft 



(4.24) 



We follow the typical practice of assuming an Arrhenius equation 
for T (Eq. (4.4)). Eq. (4.24) may then be written 



e qQ 



^ 



T EXP[E^/RT] 
y a 



(4.25) 



J 



Taking the natural log of Eq. (4.25), one has an equation of the form, 



LnT^ = e + 



io\ 



10- 



(4.26) 



which is fit, using linear regression techniques, to the experimental 



EXP 3 
points of T^ vs. 10 /T. The following analytic expression is 



KXP 
obtained for T as a function of temperature; 

Ln(Tj ^^^) = -0.747 - 0.802p|- 



(4.27) 



The actual experimental points are shown in Fig. 20 plotted as 



J 



30 



20 



10 
9 
8 

^ 7 

« 

^ 6 






ap. 



94 



-This work 
-Ref. (19) 



4.2 43 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 

IQ?^T (K") 



Fig.20. LOGjT*) vs. 10'/ (TEMPERATURE) 



95 



Log T^ vs. 10 /T. The solid line represents the linear regression 
fit given in Eq. (4.27). 

In order to use Eq. (4.24) we made the assumption that the molec- 
ular motion is isotropic. This assumption is justified on the basis 
y of corroboration from previous work (19) from which the N relaxation 

times vs. temperature were indirectly calculated. In ref. (19) the 
NH, high resolution proton (see Fig. 8) lineshape was fit to the theory 

of Pople (30). From the lineshape fitting procedure, both T^ of the 

14 

N nucleus and the indirect spin-spin coupling constant J were deter- 
mined. The work reported in ref. (19) extends from 201 to 307 K. The 
dotted line of Fig. 20 represents the results of their work over the 
normal liquid region. The correlation between our results and those 
of ref. (19) is excellent when one considers how very different the 
two methods are. With this support of our procedure, we proceed to 
'■i calculate first the reorientation activation energy, then x vs. T. 

The reorientation activation energy is calculated from the linear 
regression value of the slope of Eq. (4.26), i.e., from comparison of 
Eqs. (4.26) and (4.27) one obtains Eq. (4.28). The indicated devia- 
tion represents the statistical 99% confidence interval. 

e3 = 1.60 + 0.31 kcal/mole (4.28) 

c 

The values of T£ obtained from Eq. (4.27) are used in Eq. (4.24) to 

calculate t vs. T. We perform this calculation for: a.) The free 

2 
molecule value (e qQ/fi) = 2it x 4.08 I'lHz and b.) The quadrupole cou- 
pling constant for the NH„ molecule in solid NH„, corrected for zero 

2 
point molecular motion (14), (e qQ/fi) = lir x 3.47 MHz. The latter 

value is expected to be the' more correct, but again we let comparison 



96 



with previous work be the judge. The curves, t vs. T, obtained in 
this manner are illustrated in Fig. 21. The regions of uncertainty- 
are calciiLatad using a rather pessimistic + 10% uncertainty in the T 

1 

data. 

J Smith and Pox^7les (43) have investigated the proton relaxation 

times in pure liquid NH„ and in mixtures of pure NH with ND . By 
measuring the proton T in NH as a function of the ND_ concentration, 
they were able to separate the intermolecular and intramolecular con- 
tributions to the experimental T . They assume that the correlation 
time governing the intermolecular contribution is the same as the re- 
orientational correlation time x which governs the intramolecular 

contribution. Finally, an expression is obtained for t, in terms of 

a 

the total dipolar contribution. This expression is corrected by 

-13 
Powles and Rhodes (61) and they calculate a value of t, = 6.6 x 10 

d 

sec. at 202 K. They assume this correlation time to be identical to 
T . The value of x they calculate lies less than 5% (well within our 
experimental uncertainty) above our corresponding value of x , which 
was calculated from Eqs. (4.27) and (4.24) using the quasistatic solid 
state value of the quadrupole coupling constant (3.47 MHz.) 

We interpret their corroboration of our data as support for the 
existence of strong hydrogen bonding in liquid ammonia. If the liquid 
were not strongly bonded, one would expect the appropriate quadrupole 
coupling constant to be nearer the free molecule value than that of 
the solid. It is interesting that they interpret this same data point 
as support for the existence of little hydrogen bonding in the liquid. 
Their interpretation is based on 1^ measurements made on deuterons in 
heavy ammonia. This work was published by Powles, Rhodes, and Strange 



; 



97 



J 



X 


N 

X 


s 


s 


r- 


CO 


^ 


o 


ro 


^ 


II 


II 


< 


^ 


o 


o 


CT 


cr 


-co 


"(U 




K^-H 



o 



ro 

CM 



C\J 



CD 



CO 



CD 



lO 



ro 



in 




CM 




CVI 






UJ 


o 


a: 


C\J 


—> 


(M 


\- 




< 




a: 




til 




Q. 




IE 




III 


If) 


1- 


CM 






LU 




h- 




=> 




_l 




O 


o 


cn 


CVJ 


< 



8 



o 



in 

CD 



CVJ — - 



O 
en 



LU 
CE 

< 

or 

a. 

:e 

UJ 



> 
LU 



2 

O 



UJ 

cc 
o 
o 



(oasd) 



X 



L 



98 



(62).. From the T^ measurement and the value of t stated previously, 
they calculate the deuteron quadrupole coupling constant from an equa- 
tion of the same form as Eq. (4.24). The value they obtain for the 

2 
deuteron e qQ/h in ND (245 + 25 kHz) is close to that of a recent 

) measurement for the NH D free molecule (63), 282 + 12 kHz; but higher 

than an earlier measurement (64) for the ND- free molecule (200 + 20 
kHz). From comparison of their value with that of the NH„D free mol- 
ecule, the conclusion is drawn that little hydrogen bonding is present 
in the liquid. 

It is this author's belief that, although the Powles-Rhodes inter- 
pretation of little hydrogen bonding in liquid ammonia is completely 

J consistent with their observations, the more detailed analysis made 

possible by their work, the work of Lehrer and O'Konski (14), and our 
work strongly supports the X-ray diffraction (39) interpretation of 

strong hydrogen bonding in the liquid. Lehrer and O'Konski suggest 

14 
that the solid state shift of the N quadrupole coupling constant is 

due primarily to distortion of the nitrogen lone electron pair. The 

highly exposed position of the electron lone pair leads one to believe 

that it should be very susceptible to distortion through hydrogen 

bonding, thus providing a sensitive environmental probe. In light of 

the 33% ionic character of the N-H covalent bond and its short length 

relative to intermolecular hydrogen bond lengths, one is led to the 

J 

speculation that by far the predominant contribution to the electric 

field gradient at a deuterium (hydrogen) site is due to the electronic 

structure of the nitrogen to which it is covalent bonded. Since the 

i intramolecular N-D distance remains nearly constant in the hydrogen 

bonded solid, it seems likely that hydrogen bonding may affect the 



99 



o 1 / 

e qQ/^. only through distortion of the molecular N electronic struc- 
ture. Furthermore, the effect is quite possibly less than the 15% 

1 A 9 

reduction of the N e qQ/-!!. 

Rabideau and Waldstein (52) have measured the quadrupole coupling 
constant in ND„ from the quadrupole splitting of the Zeeman spectrum; 
they find values of 156 + 7 kHz at 75 K and 72.5 + 0.9 kHz at 159 and 
185 K. The reduction at the higher temperatures is attributed to 
molecular reorientation of the molecule about the C„ axis, but they 
assume that this motion is unlikely at 75 K. Our measurements on NH„ 

indicate a correlation time for C_ hindered rotation of approximately 

-7 

9 X 10 sec. One would not expect this correlation time for ND„ to be 

much different. Symmetry axis rotation alone may thus lead to aver- 
aging of the 200-300 kHz quadrupole coupling constant. Furthermore, 

molecular torsional oscillations will contribute additional averaging, 

2 
thus reducing the apparent e qQ/h to an even lower value. Until the 

2 
quasistatic solid state value of e qQ/h is determined for ND , an 

absolute statement of the effects of hydrogen bonding cannot be made. 

Powles and Rhodes also recognize this fact and do not support their 

conclusion with this value. 

From the information presented above, we are led to the following 
conclusions as a possible solution to the problem: 

a.) Liquid ammonia is hydrogen bonded as previously indicated 
by X-ray ar- heat of vaporization work. 

b.) The deuteron quadrupole coupling constant is not highly 
affected when hydrogen bonding occurs. 



; 



100 



4.5.2.. Interpretation of Proton T and T ^ Data from Liquid NH - 

As was mentioned in Sec. 4.5.1., Smith and Powles (43) have meas- 
ured the proton T in pure liquid NH^ and in mixtures of NH and heavy 
ammonia. Proton T^ has also been measured by Blicharski, et al (65), 
as a function of temperature. Both of the experiments were performed 
with the sample under its own vapor pressure. We repeated this work 
with a constant absolute pressure of approximately 20 p.s.i. over the 
sample. These measurements were made primarily to investigate the 
paramagnetic impurity (mainly oxygen) content of our sample. T 
measurements on protons in molecular liquids are in many cases a more 
sensitive indication of oxygen contamination than mass spectrometry 
(66). Our goal was to obtain results consistent with those of the 
Smith-Powles experiment. 

It was noticed in the course of making the T^ measurements that 
while T^ was on the order of five seconds, T^ appeared to be approx- 
imately three orders of magnitude smaller. At the time, this was 
assumed to arise from magnet inhomogeneity. In order to be certain 
that magnetic field inhomogeneity was the cause for the unusually 
large difference in the two time constants (recall that in the case of 
liquids, one usually finds T = T„) , we used the Carr-Purcell (67) 
technique to eliminate the effect of magnetic field inhomogeneity on 
!„. Our T„ measurements by this method are crude but certainly in 
error by no more than 20%. We find that indeed T„ << T , and the 
explanation of the observed temperature dependence of T„ leads to a 
quite reasonable but rough value for the proton-proton chemical ex- 
change activation energy. We now proceed to consider the effect on 
T^ and T„ of each interaction listed in the summary at the end of 



101 



Chap. II. 

EXP 

One may write a rate equation for T^ which is completely anal- 

EXP 
ogous to that of Eq. (4.2) for T , i.e., 

T. k T. 

X 1 

where the i denotes either T or T and the terms present in the sum 
are defined as follows : 



„1 _ 

dipole-dipole interaction. 



T = the contribution from the intramolecular proton-proton 



2 

T. ~ the contribution from the intramolecular proton-nitrogen 

dipole-dipole interaction. 

3 

T. = the contribution from the intermolecular proton-proton 

interaction. 

4 

T. = the contribution from the indirect spin-spin interaction. 

Intermolecular N-H dipole-dipole interactions and contributions 

through an anisotropic chemical shift tensor are neglected. The jus- 

2 
tification for neglecting the former will be seen when the T, term is 

considered. We have no information concerning the existence of an 

anisotropic chemical shift in NH„, but a calculation assuming a quite 

large anisotropy produced a negligible contribution to T.. 

The intramolecular proton-proton dipole-dipole interaction term 

T^ has been calculated for the case appropriate to NH (three spin 

1/2 nuclei arranged one on each corner of an equilateral triangle) by 

Hubbard (46) using density matrix techniques. He assumes that the 

molecule undergoes isotropic stochastic reorientation, characterized 

by the correlation time t , and obtains the following expression for 

the recovery of the nuclear magnetization along H : 



102 . 



R(t) = -^— 3J ~ = 9.192 X lO" EXP[-t/T ] + 9.908 x 10"-'-EXP[-t/T ], 

o ^ 

where, —= 4.001 x 10""'° t ' (4.30) 

b 

and -7^ = 9.585 x 10"'-° t . 

i c 

c 

The numerical values were calculated using free molecule geometrical 
parameters. Note that although the recovery is governed by two ex- 
ponentials, one approximately twice the other; it is a very good 
approximation to simply drop the first term on the right hand side of 
Eq. (4.30) because its weighting factor is two orders of magnitude 
smaller than that of the second term. In the same spirit of approxi- 
mation we approximate the weighting factor of the second term as 1, 
and the resulting recovery equation becomes: 

R(t) = EXP[9.585 X IQ-'-^x^ t]. (4.31) 

We thus have the following expression for T : 

■\ = 9.585 X 10-'-° T . (4.32) 

^1 
Richards (68) has proved that My* = R(t) for the same physical approx- 
imations used by Hubbard (46). Using these results, we have 

-y = -^ = 9.585 X 10-'-° T . . . (4.33) 
1 2 

Abragam (7) develops the theory necessary to estimate, to a good 

approximation, the effect of the intramolecular nitrogen-proton 
dipole-dipole interaction. The two nuclei are assumed rigidly fixed 
in a diatomic molecule, which undergoes random isotropic reorientation. 
Our situation is of course different from this one, but we presume 
that the relatively strong intramolecular proton-proton coupling would, 



103 



if anything, tend to reduce the effect of any nitrogen-proton coupling. 

We speculate that what we calculate here will closely approximate the 

2 2 

upper limit of T^ and T„. Abragam obtains a set of coupled differen- 
tial equations for the recovery of the respective nuclei. The equa- 
tion governing the proton M recovery is, 

-rr- = 6.03 X lO^^T (M (t) - M ) + 8.5 x 10 t (S - S ), (4.34) 
at c z o c z o 

where S (t) describes the nuclear magnetization recovery of the nitro- 
gen nucleus. The constants were calculated using free molecule geo- 

14 
metrical parameters. In Sec. 4.5.1. we presented the N relaxation 

tim.e data. On the proton relaxation time scale, ^ 10 seconds, the 

quantity (S (t) - S ) may be neglected because it approaches zero, via 

the electric quadrupole interaction, approximately three orders of 

magnitude faster than the first term. Thus, in this approxim.ation, 

~ = 6.03 x 10^ T . (4.35) 

T^ '^ 

The transverse relaxation time is calculated from Eq. (89), p. 296 of 

Abragam. It is 

-— = 2.83 X 10^ T . (4.36) 

T ^ 

^2 

To calculate the intermolecular proton-proton contributions, we 

appeal to additional work by Hubbard (69) . In this paper Hubbard 

3 3 

calculates T^ and T„ assuming: a.) The short correlation time limit 

2 
holds, i.e., (o) t ) << 1, b.) The spin 1/2 nuclei are all at equi- 
valent positions at a distance b from the center of a spherical mole- 
cule of radius a, and 3.) The motions of the molecules can be con- 
sidered to be rotation' and diffusion, characterized by a single cor- 

2 
relation time % = 2a /D, where D is the translational diffusion 



104 



coefficient. The resulting equation is 



^ ^"^^ {1 + 0.233. 



T,3 T,3 5aD 
^1 2 



a 



+ 0.15 



+ •••}, (4.37) 



where ri is the number of spins (protons in our case) per unit volume. 

The use of this formulation in the case of NH- is indeed question- 
able. One is concerned in particular with the use of a single cor- 
relation time to describe both reorientation and diffusion processes. 
Perhaps a much better model for liquid NH_ would be one in which a 
single correlation time describes only rotation about the symmetry 
axis and another distinct correlation time describes both diffusion 
and reorientation about axes perpendicular to the symmetry axis . We 
will nevertheless use the Hubbard model as an approximation. 

For the constant b in Eq. (4.37), we choose the distance from the 
molecular center of mass to a proton. Using free molecule geometry, 
we obtain a value of 0.985 % for b. The effective molecular radius, 
a, which we use, is one-half the mean distance between nearest neigh- 
bors, 3.56 A, determined from X-ray diffraction work (43); i.e., 
a = 1.78 A. Substituting these parameters into Eq. (4.37), one 

obtains , 

1 1 1.60 x 10"^ ,, „, 

-3=- = ^ . _ (4.38) 

1 2 

The self -diffusion constant D has been measured over the normal liquid 

range by McCall , Douglass, and Anderson (70) and we will subsequently 

3 3 

use their .-alues to calculate the T^ and T^ contributions from 

Eq. (4.38). 

We appeal to Abragam (7), pp. 305-313 to evaluate the scalar re- 

4 4 
laxation contributions, T and T„. Recall that' the observable in- 



105. 



direct spin-spin interaction, discussed in Sec. 2.3.4., is a scalar 
interaction of the form hJI"S between the nitrogen nucleus and protons 
on the same molecule. Due to the scalar nature of the interaction it 
is invariant to molecular reorientation. We discussed qualitatively 
in Sec. 3.4.2. the effect this interaction could have on the FID; now 
we must consider how Zeeman transition inducing fluctuations may be 
transmitted to proton sites via this interaction. Fluctuations at 
proton sites may be produced by the indirect spin-spin interaction 
through two distinct physical processes: a.) Chemical exchange of 
protons between neighboring molecules, and b.) Rapid fluctuation of 
the nitrogen nucleus among its Zeeman states due to electric quadrupole 
moment coupling to the rapidly reorienting molecular electric field 
gradient. Magnetic field fluctuations by process a.) occur when the 

N nuclei on two molecules are not in the same state when chemical 
exchange of protons occurs between them. Abragam labels relaxation by 
processes a.) and b.) respectively, scalar relaxation of the first 
kind and scalar relaxation of the second kind . 

Scalar relaxation of the first kind is only effective in causing 
Zeeman transitions when x is within a couple of orders of magnitude 
of the Larmor period. When this process is effective the proton mul- 
tiplet structure is of course not observable, i.e., no beat pattern is 

observable in the FID although T„ may be many times J . The follow- 

4 4 

ing equations relating T and T„ to t are derived with the assimiption 

that when chemical exchange occurs it is. an instantaneous process. We 

have, 

1 1 J^S(S + 1) le , . 

1 I S e 



106 



and 

^2 



1 J^S(S + 1) 



T 
T + 






(4.40) 



14 
where S = 1, the N spin, and ca and co are respectively the Larmor 

precession frequencies of the protons and the nitrogen nuclei. 

Scalar relaxation of the second kind is possible when the S nucle- 

14 

us ( N in our case) has an independent relaxation mechanism which 

limits its T^ to less than J , e.g. electric quadrupolar coupled with 
molecular reorientation. Again, the multiplet structure is destroyed 

under these circumstances. In the short correlation time limit, where 

14 14 4 

T^ = T„ for the N nucleuss, the equations relating T of N to T^ 

and T„ of the protons are identical with Eqs. (4.39) and (4.40), but 

14 
with T replaced by T of N. Our nitrogen T measurements, when 

used in Pople's theory (30), and high resolution CWNMR work (19,71,72), 

indicate that in pure NH„ the multiplet structure is not destroyed. 

Therefore, we can expect no contribution from scalar relaxation of 

the second kind to proton T (substitution of nitrogen T^ values for 

4 14 
T in Eq. (4.39) yields values on the order of T ^ 10 seconds.) 

Scalar relaxation of the second kind has a drastic effect on proton 

T„, as was discussed in Sec. 3.4.2. Eq. (4.40) with x replaced by 

the nitrogen spin-latt'ice relaxation time is not even approximately 

, . , , ^nitrogen ^-1 , , ^, ^ ^ • ^ 

valid when T^ < J ; whereas under the same set of circiimstances 

Eq. (4.39) is a good approximation. This is a manifestation of the 
widely separated regions of frequency to which the respective relaxa- 
tion times are sensitive. The more exact theory of Pople (30) is 

4 
applicable for predicting T^ under the circumstances. We note from 

previous high resolution work on pure NH (19,71,72) that the multi- 



107 

plet structure is indeed observable, but the lines are broadened, 
through the indirect spin-spin interaction, by electric quadrupole 
induced fluctuations among the nitrogen Zeeman levels. Recall that 
two distinct transverse relaxation times govern, the decay of M *(t). 
One may estimate from the high resolution traces of ref. (19) that 
T^ is on the order of 100 msec, roughly two orders of magnitude 
less than Tf^ 

Although the inhomogeneity of our magnetic field limited the actual 
duration of the proton FID to about 4 msec, a far too short FID to 
observe the beats predicted in Sec. 3.4.2., by use of the Hahn-Max^^/ell 
spin echo technique (16) we increased our pulse spectrometer resolution 
to the degree such that a J splitting in NH_ as small as 10 Hz could 
be observed. We did not, however, observe the expected beats in the 
echo envelope. This was indeed a disturbing dilemma until a search of 
the literature revealed an early paper by Ogg (72) describing his 
difficulty in observing, by high resolution CWNIffi techniques, the ex- 
pected NH„ proton multiplet structure. He found that an H^O impurity 
of only a few parts per million in his NH samples completely x^7ashed 
out the multiplet structure and narrowed the observed singlet to a 
width less than the center line of the pure sample multiplet. This 
experiment indicates that H„0 is a fantastically efficient catalyst 
for proton-proton chemical exchange. If one could arbitrarily vary 
the H2O concentration in liquid NHo while observing the true FID, one 
would observe the following effects as the H„0 concentration was in- 
creased: a.) Pure NH„ — a FID, xjith beats of frequency J, decaying 
with time constant of roughly 100 msec, b.) NH + critical concentra- 
■ tion of H„0 — a FID with no observable beats and T„ on the order of 



108 



J = 23. msec, finally, c.) NH» + high concentration of H2O— a FID 
with no beats and T„ = T , where T may be affected by the exchange 
process according to Eq. (4.39). Our T measurements using the un- 
modified Carr-Purcell (67) pulse sequence are plotted as a function of 
absolute temperature in Fig. 22. On the basis of our T^ measurements 
and the fact that we could find no indication of multiplet splitting 
in our sample, we presume that the theory presented for scalar relaxa- 
tion of the first kind is applicable for our sample. Before continu- 

-2 
ine. we note that since our T„ measurements indicate t -^ 10 sec, 
=>» 2. e 

TI^ '-J 10 sec, we may neglect the contribution of Eq. (4.39) and the 

second term in brackets in Eq. (4.40). We must therefore contend only 

EXP 
with the following contribution to !„ : 

-V = 1.27 X 10^ T . (4.41) 

^2 

, . r^EXP , ^EXP 
We have now considered all contributions to T and T„ , and 

combine Eqs. (4.33), (4.35), (4.36), (4.38) and (4.41) to write the 
theoretical expressions. We denote by a subscript t that these values 
are theoretical. One obtains, 

-^=1.02x10^^ +1.60.JLi0l, (4.42) 

EXP X.U^ A XU L ^ 

T 
It 



and 



^ 9.87 xlO^Q T + ^-^Q I ^°"% 1.27 X 10^ T . (4.43) 



3XP c D e 

^2t 

In Sec. 4.5.1. we calculated the rotational correlation time vs. 
temperature and corroborated our results with those obtained through 

alternate approaches. We thus feel that the values calculated for x 

14 
using the solid state shifted value of the N quadrupole coupling 



109 




O 

(M 



in 
ro 



O 

to 



in 
cvj 
eg 



LU 
O 3 



OJ 



< 
ir 

LU 



Q. 


ID 


:e 


OC 


LU 


3 


tnl- 


H 


^UJ 


^ 


1- 
_j 


Hi 
S 


o 


u 


en 


t- 


ca 




(M 





o 


o 


o 


o 


O 


o 


o 


o 


O 


o 


O) 


CO 


N- 


U) 


in 


^ 


ro 


OJ 



m 
O 



O 

o 

CVJ 



15 



o 

CD 



(oesuj) \ 



H- 



LU 



s 



LlI 

q: 

8) 

z 

8> 

z 
o 

CE 

a. 



if 



no 



constant (3.47 MHz) may be used for t in Eqs . (4.42) and (4.43) with- 
out further justification. The values of the diffusion constant D 

were taken from ref. (70). Fig. 23 depicts a comparison of our ex- 

EXP 
perimental results with those of Smith and Powles (43) and the T^^ 

values calculated from Eq. (4.42). The top, solid curve is the cal- 

EXP 
culated T., values. The heavy dotted straight line represents the 

Smith-Powles results, as well as could be determined from Fig. 1 of 
ref. (43). The actual experimental scatter of our points, taken at 
nine different temperatures, is represented as heavy vertical bars 
through the thin dotted line, which represents a smooth curve draxra 
to minimize the experimental deviations. Note that the theoretical 
values predict a gentle curve, but nothing so striking as what our 
data display. One might easily explain a slight difference in the 
absolute value of different experimental results in terms of slight 
differences in sample purity, but the large curvature of our values 
is unexplainable in these terms. We note that the slope of our ex- 
perimental curve is approximately equal to that of the other curves 
at the low temperature end but begins to rapidly decrease when the 
temperature reaches approximately 210 K. This interesting behavior 
has not been satisfactorily explained; possibly a clue could be ob- 
tained through an isothermal measurement of T vs. sam.ple pressure. 
In view of the critical dependence of T on sample purity, pressure 
work of the type proposed would be very difficult. 

We now return to Eq. (4.43) and our experimental measurement of 

T^ . Simplification of Eq. (4,43) may be achieved by noting that 

FXP 
1/T --^ 10 and the sum of the first two terms on the right hand side 

are on the order of 0.1, a contribution less than the estimated +20% 



Ill 



_: J 1 1- 



30 



20 



10 

9 

8 
7 



o _ 

uj 6 



K 5 



4 



BP 




Theory 
Ref.(34) 

This work 



42 4.3 44 4.5 4.6 4.7 4.8 4.9 5.0 5.1 



Fig.23.L0Gn. ) vs. lOy(TEMPERATURE) 



112 



experimental error. If these terms are dropped and one assumes the 

, standard Arrhenius form for the temperature dependence of x^, 

Eq. (4.43) may be written in the familiar form, 

,e 



LnT^^ =A4- 



i 



io\ 



^. C4.44) 



suitable for fitting experimental data by linear regression. We use 

•pyp EXP 

our experimental values T^ for T^^ and fit to Eq. (4.44). Fig. 22 
depicts our data, represented by the circles, with the solid line rep- 
resenting T„ . One obtains from the linear regression fit a value 



of. 



E^ = 0.99 + 0.32 kcal/mole, (4.45) 



for the proton-proton chemical exchange activation energy. The error 
limits represent a linear regression estimate of the 80% confidence 
interval. This value is considerably lower than one might expect from 
previous work (19) , which employs KNH rather than HO as a catalyst 
for proton-proton exchange. They find the chemical exchange activation 
energy to be independent of BCNH„ concentration and equal to 4.0 + 0.5 
kcal/mole. It is conceivable that the proton-proton exchange activa- 
tion energy may be a function of catalyst. In light of the tremendous 
efficiency with which H„0 impurity in NH destroys the pro.ton multi- 
plet structure, Eq. (4.43) may well be a reasonable estimate. 



CHAPTER V 
INSTRUMENTATION AND MEASUREMENT PROCEDURE 

5.1. Design of Cryostat and Sample Handling Procedure 

5.1.1. Sample History and Handling; Procedure 

The ammonia sample used for all experiments came from, a single 
source: The Matheson Co. It was analyzed by J. Catesel of the 
Matheson Co. and found to be 99.999% pure NH . A first prerequisite 
for design of the sample handling system was the preservation of sample 
purity. In particular, oxygen contamination was to be avoided. The 
handling of ammonia presented somewhat of a problem in that it is quite 
toxic and chemically active. Dr. J. Kronsbein (73) advised the use of 
only ferrous metals for handling the gas. The sample chamber required 
use of a nonconductive, nonmagnetic material. We chose the polymer of 
trlfluorochloroethylene, known by the trade name of Kel-F, for the 
sample chamber primarily for its inertness and desirable low tempera- 
ture properties. 

A diagram of the gas handling system is shot-m in Fig. 24. It is 
constructed of only ferrous metals, teflon and Kel-F. All joints are 
either welded, metal-to-metal sealed, or sealed using teflon. The 
system xjas thoroughly cleaned, assembled, and checked for leaks using 
a Veeco MS9AB mass spectrometer leak detector. The complete system 
was then connected to a standard design cold-trapped diffusion vacuum 
pump and pumped out to roughly 10 Torr. Outgassing was accomplished 

113 



114 



Valves 



To Vacuum System 




Steel Tube 




99.999 7o Pure NH, 



400 in^ Low Pressure 
Storage Cylinder 





•^ 



cr^ 



Stainless 
Steel Tube 



Cryostat 




Kei-F 
Sample 
Chamber 



Fig. 24. DIAGRAM OF THE GAS HANDLING SYSTEM 



115 

by heating the system to above 370 K using a heat gun. This process 
was repeated until no increase in pressure was observed upon heating. 
The pump-out procedure was carried out over a period of several days. 
Ammonia gas was introduced into the system at a pressure such that 

when the sample chamber was filled with condensed liquid NH„, a pres- 

2 

sure of approximately 5 lb/in above atmospheric remained in the low 

pressure system. The sample system circuit remained intact throughout 
the experiment. 

The first sample remained in the cryostat and low pressure system 
from 10 May 1968 until 10 June 1958. During the course of this period, 
the sample was condensed into the sample chamber and boiled off a num- 
ber of times. The first sample was then pumped out and the system 
outgassed, and a second sample was introduced and condensed into the 
sample chamber. Six runs were made on Sample II, measuring proton T.. 
vs. temperature in the liquid. Consistent results were obtained, but 
the value of T^ at 195 K was consistently about 1.5 sec lower than 
that of ref. (43), indicating a slight impurity. The sample was 
changed again and further proton T^ measurements made. It was found 
that if the sample remained in the liquid state over a period of time 
(roughly three days), the proton T values increased slightly, reach- 
ing a limiting value of — 5 sec at 198 K. A temperature run was made 
on sample III (proton T vs. temperature) after the limiting value 
\ was reached, then the sample was again changed. Proton T values 

remained invariant, within experimental error of approximately +10%, 
to both the amount of time the sample remained in the liquid state 
and a change of the external magnetic field. The proton T data in 
solid NH were taken using sample IV, then confirmed by changing 



116 

samples again and taking spot points throughout the temperature range. 
Perfectly consistent results were obtained from both samples in the 
solid. 

5.1.2. Construction and Performance of the Temperature Control Cryostat 

The temperature control cryostat was designed to be as versatile as 
possible, and is useful in the temperature region from 4.2 to 200 K. 
The design is such that complete disassembly is easily effected in 
order that repairs or modifications may be made with a minimum of dox.m 
time. It would be quite feasible to design a number of interchangea- 
ble probe sections for use in different experiments. This disserta- 
tion is obviously no place for detailed design drawings, but Fig. 25 
will Illustrate the basic design details. 

The control section is machined from a solid piece of aluminum. 
All tubes connecting the probe to the control section are ten mil wall, 
nonmagnetic stainless steel. The vacuum space between the two outer 
tubes prevents the sample from solidifying and plugging the sample 
tube. He exchange gas occupies the space between the sample tube 
and inner vacuum jacket tube. The top of the brass outer can is seal- 
ed with a Wood's metal solder joint. The temperature bomb was ma- 
chined from a single cylinder of oxygen-free high-conductivity copper; 
its final mass is 909 grams. Six 1/4 x 20 brass bolts fasten the 
temperature bomb, and a vacuum tight seal is effected using an indium 
0-ring. A three mil brass shimstock radiation shield radiatively 
decouples the temperature bomb from the outer brass can. Convective 

heat transfer betx^?een the temperature bornh and the outer can may be 

4 
controlled at will by varying the pressure of a He exchange gas in- 
side the outer can. Heat transfer betv7een the sample and the 



117 



37" 30.1" lir 



i3.r 







ya'od - io MIL U.PIUI. 



i/Z* oj ' to Mil. Viftui. OT^^/VtffiS 

/«/« lAttiuM Jacket 'TZ^e 
o 

OoTSIOC. lAiUWr JicUiT TuoS 






^<uuM JrtCKir 



3 Mil. ^/fflJS ^r^'^JT-oc*; 
Bkasj Vacuum Jfi^CK^r C.Ar 



/r /Pis/trAf^a /He/ii<'K^fi£Teix 



-OFHC Ca/vsn Bom a 



Fig. 25. THE TEMPERATURE CONTROL CRYOSTAT 



118 



temperature bomb is maintained both by thermally grounding the coil 

4 

surrounding the sample chamber and by providing a He exchange gas. 

Temperature control is effected by first surrounding the outer can 

with a slowly boiling cryogenic liquid, and then allowing heat trans- 

4 

fer between the temperature bomb and the cryogenic bath, via a He 

exchange gas, until the desired sample temperature is reached. One 

-3 
then evacuates the outer can to a pressure ^ 10 Torr and supplies 

energy to the temperature bomb, via a heater, at the same rate that 
energy is being lost to the cryogenic bath. The bomb heater was 
wound in two sections, each covering approximately one inch of the 
bomb length, positioned in such a manner that the center of the sample 
chamber lies between the coils. The coils were wound independently 
so that a thermal gradient over the sample could be cancelled by 
varying the relative power input to the coils (see Fig. 28). Each of 
the coils consisted of 700 Ohms of teflon coated, copper-nickel alloy 
wound noninductively around the bomb. 

Electrical leads from the control head into the probe section are 
as follows: 1.) Three //32 varnished copper wire leads for the heat- 
ers — electrical resistance of each is 0.6 Ohms, 2.) Four #36 varnish- 
ed copper wire leads for the platinum resistance thermometer — elec- 
trical resistance of each is 1.4 Q, 3.) Tv70 three mil copper wires 
for a thermocouple pair and 4.) The rf lead of rg/u 178b shielded 
50 Ohm cable. The rf lead was passed between the 1/8 inch stainless 
sample tube and the inner vacuum jacket tube; the shield was grounded 
to the temperature bomb. All remaining electrical leads were chan- 
nelled do^ra a 1/4 inch id. stainless steel tube and thermally grounded 
to the outer brass can. Apiezon "N" grease was used to provide good 



119 



thermal grounding (74) between the heater and the temperature bomb, 
and the electrical leads and the brass can. 

5.1.3. Temperature Measurement, Contr o l and Stability 

Temperature measurement and control were effected through the use 
of a Leeds & Northrup (L & N) type 8164 four-lead platinum resistance 
thermometer which was calibrated by the National Bureau of Standards 
(Test #037794). The actual thermometer used was L & N serial #1692601. 
Through use of an L & N type G-1 Mueller Bridge and a type 8068 mer- 
cury commutator, the absolute thermometer resistance was measured 
irrespective of lead resistance. We used a Keithley model 149 milli- 
microvoltmeter as a null detector. 

The thermometer, a cylindrical device 6 cm long with an o.d. of 
5.5 mm, was mounted in a hole drilled into the side of the bomb par- 
allel to its axis. Care was taken to coat the thermometer over its 
complete length with Apiezon "N" grease in order that good thermal 
contact was established. Care was also necessary when soldering the 
thermometer leads as it is advantageous to minimize both thermal emf, 
caused by poor solder joints, and the difference in lead resistance of 
the two lead pairs. Good quality tin-lead solder was found to be much 
more satisfactory than a "low thermal emf ", solder (this perhaps re- 
flects the author's inability to obtain a decent solder joint with the 

latter). 

Thermo-zeter resistance decreases from 25.4800 Abs. Ohms at the 
triple point of water (273.16 K) to 1.91969 Abs. Ohms at 50 K. Fur- 
thermore, the resolution (dR/dT) decreases from 0.1 Ohm/K at 273.15 K 
to 0.094 Ohm/K' at 50 K. The resolution becomes very poor at tempera- 
tures below 20 K, and the thermometer is almost unusable below 10 K. 



120 



Throughout the region of temperature in which measurements were made, 
we could easily resolve temperature changes of +5 mK corresponding to 
a resistance change of 5 x 10~ Ohms. We estimate our absolute tem- 
peratures to be accurate to +0.1 K and our temperature control to be 
stable to better than one part in ten thousand. 

The operational principle of our temperature control unit is 
straightforward: For a given set of conditions (cryogenic bath tem- 
perature, desired sample temperature, pressure in cryostat vacuum 
jacket, etc.), the power input to the bomb must equal the energy lost 
per unit time to the cryogenic bath, if one expects an equilibrium 
situation at a desired temperature. Operationally, one sets the 
Mueller bridge to the resistance corresponding to the desired bomb 
temperature; if the bridge is out of balance an error voltage is in- 
dicated by the Keithley model 149. The magnitude of the error voltage 
is a function of the amount of bridge out-of-balance and the voltage 
gain setting of the Keithley. The polarity of the error voltage is 
determined by whether the bomb is too hot or too cold relative to the 
desired temperature. If the bomb is too cool, one increases the power 
input until a steady state condition at the desired temperature is 
reached; if too hot, the power input is decreased. The power input is 
a quadratic function of the voltage applied across the bom:b heaters. 
A perfect steady state condition at a constant input power cannot be 
conveniently met because slowly changing conditions of the system 
(vacuum fluctuations, power supply drift, etc.) continuously demand 
slightly different input power. We provide this slight variation in 
input power by applying the error voltage developed by the Keithley to 
a programmable pov;er supply. The error voltage developed (-10 to to 



121 



+10 Vdc) produces roughly an equal change in voltage across the bomb 
heater. Thus the change in power input is proportional to the square 
of the out-of-balance signal. This quadratic behavior provides ex- 
ceptionally good temperature control. 

The temperature control loop is illustrated in Fig. 26. All con- 
trol is derived from a programmable Krohn-Hite model UHR-T361R power 
supply. A Heathkit model PS-4 regulated power supply is not a part of 
the control loop and is used as a booster to facilitate rapid changes 
of the bomb temperature. The gain of the loop is determined by the 
sensitivity setting of the Keithley. The temperature stability fig- 
ures presented earlier are based on a sensitivity of 1 yV full scale 
with 6 V applied to the Mueller bridge. No loop instability was ever 
observed; this was no doubt due to the low thermal resistivity and 
high thermal capacity of the bomb, and the relatively small available 
power change. 

The average power input to hold a given temperature under typical 
conditions is of some interest both because it is a measure of the 
power input to the cryogenic bath and because it facilitates setting 
the average heater bias voltage. Fig. 27 illustrates the average 
power input to hold a constant temperature vs. the absolute tempera- 
ture, for the case of a slowly boiling liquid nitrogen bath. The 

-4 
pressure in the cryostat vacuum jacket was approximately 10 Torr 

and was held constant by continuous pumping. The two heaters were 

connected in series for this measurement. Notice that a +0-07 watt 

control, corresponding to our available +10 V control, has a steady 

state effect of +12.5 K at 90 K but only a + 7.25 K effect at 200 K. 

We mentioned previously that the heater coils were wound such that 



122 



Heothkit PS- 4 



Krohn-Hite UHR-T36IR 




Keithley model 149 
Milli-microvoltmeter 




LSN Type G-l 
Mueller Bridge 



>,^^ 



L&N Type 8154 
R. Thermometer 



OFHC Copper 
Temperoture Bomb' 



Heoter Coils 



J 



■A. 



Fig. 26. BLOCK DIAGRAM OF THE TEMPERATURE CONTROL LOOP 



123 




CD 



CO 



f~ tD ih ■^_ to 
(SliVM) indN! yBMOd 



cvj — 



^ 



en 



124. 



any temperature gradient existing in the sample chamber could be can- 
celled. In order to determ.ine if this procedure would be necessary, 
the temperature gradient was measured between the top and bottom of 
both the sample chamber and the outside of the temperature bomb, for 
the case of a steady state temperature established by a current flow- 
ing through the series connected heaters. The measurements were made 
using a gold~2.1 atomic % cobalt— copper thermocouple pair placed in 
the positions a-a and b-b, as depicted in Fig. 28. The outside 




Stoinless Sample Tube 



R. Thermometer Hole 



-Heater Coil 






■Kel-F Sample Chomber 



OFHC Copper Bomb 



He Exchange Gas 



k 



l.so 



Fig. 28. SAMPLE CHAMBER 8 TEMPERATURE BOMB 



125 



thermocouple pair (a-a) was thermally grounded to the respective ends 

of the bomb via mica insulation and Apiezon "N" grease; measurements 

-4 
were made with the vacuum jacket evacuated to roughly 10 Torr. The 

inside thermocouple pair was attached to copper disks, 60 mils thick 

and 450 mils in diameter, separated by a two inch styrofoam spacer and 

positioned inside the bomb as shown in Fig. 28. Throughout the meas- 

4 

urements, a He pressure of one atm. was present inside the bomb. 

Thermopotential was measured in each case after thermal equilibrium 
was established relative to a liquid nitrogen bath. The actual meas- 
urements were m.ade with the Keithley model 149 milli-microvoltmeter . 
In order that any extraneous thermopotential generated at the Keithley 
input be cancelled, a thermal free reversing switch was used; the 
average of the forward and backward voltages was taken as the true 
value. Fig. 29 illustrates the results of the measurements plotted 
as the difference in temperature of the respective thermocouples vs. 
the steady state temperature. The thermocouple calibration tables 
used vrere those of Powell, et al (75). We consider these results to 
be excellent justification to ignore the problem of thermal gradients 
across our sample when the heaters are connected in series; thus the 
series connection was used for all experiments. 

5.2. Measurement Procedures Used 

5.2.1. Measure ment of T by 90°-t-90° Pulse Sequence 

This measurement procedure is quite straightforx>7ard for the case 
of the existence of a single spin-lattice relaxation time. We have 
illustrated from various theoretical results that the existence of 
a single T^ governing the relaxation of protons in NH„ is a good 



126 



ACROSS SAMPLE (K) 




oq 

<M 
CJ 



( y ) 8W0a JO H01H31X3 



o 




o 




CJ 






til 




q: 


o 

CO 


3 
t- 
< 




cr 




UJ 


UJ 


0. 


oi 

log 


5 

UJ 

cc 


0- 


UJ 


s 


m 


UJ 


5 


1- 


< 




X 


2h 


o 


2:5 


UJ 


_j 


_l 


o 


a 


CO 


5 


CD 


< 


< 


(rt 


o 


(/> 


OJ 


> 




Ui 




o 




2 




UJ 




(T 


o 


UJ 


o 


U. 




U. 




O 




UJ 




tE 




U 




1- 


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CD 


cr 




Ul 








2 




UJ 




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&> 




CVJ 




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127 



approximation over most of the region studied. If one makes this as- 
sumption, Eq. (3.69) may be used to measure T- . The initial conditions 
are established by applying a 90° pulse, i.e., M (t=0) = 0. After the 
90° pulse one observes a FID, the height of which, immediately follow- 
ing the pulse, is proportional to the thermal equilibrium magnetiza- 
tion M . In practice however one cannot observe the FID at t=0 due to 
o 

the intrinsic deadtime of the electronics. A long deadtime may com- 
pletely obscure a short FID; we consider this problem in more detail 
later. In practice one chooses a point at time t from the initiation 
of the 90° pulse (a point such that the FID is observable) , and makes 
all amplitude measurements at this point, keeping t constant. If one 
applies a second 90° pulse a time t < T after the first, the ampli- 
tude at t + T will be smaller because thermal equilibrium has not 
o 

been re-established. We associate the amplitude V, (0) obtained in 
this manner after the first pulse, with Mq ; likewise, if the 90° pulses 
are t apart, we associate V (t) with M2(t). One may then write 
Eq. (3.69) as, 

Vd(T) = Vd(0)[l - EXP(-t/T^)]. (5.1) 

The two 90° pulses, a time t apart, are referred to as a single 
90°-T-90° pulse sequence. 

If several 90°-t^-90° pulse sequences are applied with t different 
in each case, one may plot the recovery envelope if the time between 
each pulse sequence is greater than 5 T . Fig. 30 illustrates the 
procedure described above. 

The spin lattice relaxation time T^ is extracted' from the experi- 
mental data using Eq. (5.1). One plots Ln[l - V^^ (t)/V^ (0) ] vs. t 



128. 



and finds 



T^ = - 1/m, 



(5.2) 



where m is the slope of the straight line. This procedure has the 
advantage of illustrating immediately: (1.) The precision of the 
90° pulse tuning and (2.) any non-linearity in the recovery. If the 
90° pulse is tuned correctly the straight line should pass through the 
origin, if not the intercept will be finite. One should be particu- 
larly dubious of any T value obtained when the straight line inter- 
cept is positive. 

Recovery Envelope 
'90' Pulse 



— H H — Amplifier Dead Time 
-90' Pulse 




TIME 



Fig. 30. THE 90*- 1 - 90' PULSE SEQUENCE 



Another procedure used to obtain T, is that whereby one simply 

seeks the pulse separation t which makes the quantity Ln(l - V,(t)/V, (0)) 

d d 

equal to minus one. At this point, clearly '^ equals T^ . This proce- 
dure may lead to large errors if the 90° pulse tuning is not correct. 

The procedure used in this work is the linear regression technique. 
The experimental data were fit to a straight line of the form, 

Ln(l - V,(t)/V^(0)) = mi + b. (5.3) 

d d 



129. 



This procedure has the advantage of removing random errors in a sys- 
tematic manner. One may also obtain an estimate of the variance of 
the linear regression coefficient m from which one may estimate the 
statistical interval of confidence of T^ . The intercept b, from 
which one may estimate the accuracy of one's 90" pulses, is also 
available. This procedure is of course entirely too laborious with- 
out the assistance of a small computer. We found the Hewlett-Packard 
9100A ideal for this purpose. Clearly this procedure would be xvell 
suited for an on-line computer fed by a boxcar integrator. All sta- 
tistical procedures used were taken from Ostle's book (76) on the 
subject. 

The actual data were taken using the beam intensification feature 
of the Tektronix 545 oscilloscope. This procedure is identical to 
that explained by De Reggi (77) and thus will not be reiterated here. 

5.2.2. The Measurement of the Spin-Spin Relaxation Time 

As motivation for using these techniques, we first discuss the 
effect of magnet inhomogeneity on T„ of liquids. In Sec. 3.3.2. we 
considered the relationship between the lineshape and the FID when 
our external field H was perfectly homogeneous, but this is never the 
case. Inhomogeneities in real magnets are primarily the result of two 
factors: 1.) The finite diameter of the pole faces leads to a radi- 
al field gradient; the field is largest at the center and decreases 
as a function of radial distance. 2.) A very serious form of in- 
homogeneity results from different regions of the pole faces magnetiz- 
ing to different values of H for a given constant magnetizing force. 
The latter problem should not occur in a well-designed magnet, but 
the first is an intrinsic property. One widely used scheme of reducing 



130 

the radial gradient over an area at the center of the pole faces is to 
provide a ridge of increased thickness around the outside edge. The 
width of the raised ridge and its thickness are used as parameters to 
adjust the homogeneity of the central portion of the field. The pole 
faces on our magnet are not of this design and the radial gradient is 
approximately 50 mG/cm at H = 7.05 kG over a circular area roughly 
two centimeters in diameter. One may, incidentally, improve the ef- 
fective homogeneity of a flat pole face magnet by the following very 
simple operation: Run the field approximately 100 G above the desired 
field, then reduce it to the desired value. 

For protons, a difference in local field of 50 mG at 7.05 kG cor- 
responds to a difference in precession frequency of roughly 2.1 kHz. 
Since the natural linewidth of most liquids is in the range of 0.1 to 
10 Hz, clearly the lineshape is characteristic only of the magnet. 
We wish to consider the effect of the magnetic field inhomogeneity 
and sample geometry on the FID, for the limiting case where the natu- 
ral lineshape f(v) may be approximated by a delta function. In this 
case, as we showed in Sec. 3.4.2., the magnetic field distribution 
function over the sample takes the place of the lineshape function 
in Eq. (3.29) and we have 

00 

V,(t) = a^ /g(v)cos27TVtdv. (5.4) 

—CO 

The magnetic field distribution function g(v) is written in terms of 
the distribution of Larmor frequencies it produces over the spin en- 
semble, as seen in a frame rotating at v . The problem is simplified 
greatly by choosing one's laboratory coordinate frame and sample ge- 
ometry in such a manner as to maximize symmetry of the problem. In 



131. 

practice this is usually done by simply centering a symmetric sample 
chamber over the center of the magnet pole faces. 

We consider the experimental case that we were presented with: 
That of a long, thin cylindrical sample chamber, of length i and 
radius a, geometrically centered relative to the magnet pole faces. 
We use the coordinate system shown in Fig. 31(a.); and since a << Ji 
we may approximate the magnetic field seen by the sample as 

H = k(H^ - G|y|), (5.5) 

where G is the radial field gradient constant. Note from Eq. (5.5) 
that both +y halves of the sample 'see' identical field distributions. 
If we choose v = yH /2it, g(v) becomes 



g(v) = 



4tt yG& 
yG2, ^ ^ 4tt 

(5.6) 

v>^ 

4Tr 



Substitution into Eq. (5.4) gives, 

V^(t) = a^ sin(y £Gt/2) ^ ^^^^^ 

We plot values calculated from Eq. (5.7) in Fig. 31 (b.) using ' 
G = 8 mG/cm, £ = 5 cm, y - 2Tr x 4.26 x 10 and a„ = 1. For compari- 
son, a FID from our second NH„ sample is normalized and plotted on 
the same graph. Clearly the experimental FID is more highly damped 
than the theoretical curve, no doubt due to the indirect spin-spin 
interaction. The important point is that the oscillation is present, 
and explainable in terms of a not unreasonable gradient across the 
sample chamber. We used the technique mentioned previously to maxi- 
mize the field homogeneity. 



132 



/Sample Chamber 



Mognet Pole Face 




Rg. 3l(a.). SAMPLE CHAMBER 8 MAGNET COORDINATE SYSTEM 




Experimental points 



Absolute value of Eq.(5,7) 



2 4 6 



^^^c 



10 12 14 
t (msec) 




18 20 



Fig. 3Ub}. RELATIVE FID AMPLITUDE! PROTONS NH^) vs. TIME 



133 



In the literature one finds mentioned an apparent transverse relax- 
ation time T„. T' is the effective transverse relaxation time that 
one associates with the observed decay of the FID after a single 90° 
pulse. In the case of an oscillatory decay one might define T„ in 
terms of the envelope of the oscillatory maxima. This quantity is 
tenuous, at best, but useful, primarily as a parameter by which to 
gauge the homogeneity of one's magnetic field. For the case of most 
solids, T„ is meaningless because the dipcle coupling in the solid 
causes such rapid damping of the FID that one observes essentially the 
true T„, assioming that a magnet of reasonably good quality is used. 

In light of the preceding analysis one could wonder how the 
intrinsic liquid sample T2 may be measured when one is forced to work 
with real magnets. The elegant spin-echo technique may be used to 
eliminate the effects of magnetic field inhomogeneity on measurements 
of T„ . Hahn introduced the spin-echo technique in a classic, very 
lucid paper (78) concerning the response of the spin system to a 
90''-T-90'' pulse sequence, when T* < t < T^. He finds that, although 
the spins fan out rapidly in an inhomogeneous field after a 90° pulse, 
the spin ensemble may be refocused by a second 90° pulse at a time x, 
subsequent to the first, if each spin 'sees' relatively the same 
field at times t=0 and x. If x meets this condition, the refocused 
spin ensemble creates a maximum in the nuclear signal at a time t 
after the second 90° pulse. This maximum is called a 'spin-echo.' 
The spin echo is maximized by a 90°-x-180° pulse sequence as is ex- 
plained in the very readable Carr-Purcell paper (67).. Any process 
which tends to change the local nuclear fields over the period x will 
reduce the echo amplitude. The process of self-diffusion in an 



134 



inhomogeneous magnetic field is particularly effective in reducing the 

echo amplitude. Hahn finds that the echo envelope, obtained by plot- 

3 

ting echo maxima for different x, is modulated by the factor EXP(-t k) , 

where k is proportional to the product of the square of the field gra- 
dient with the molecular self-diffusion constant. 

The Carr-Purcell technique tends to eliminate the damping effect 
of diffusion through an inhomogeneous field by applying a series of 
180° pulses after the single 90° pulse (see Fig. 34). They find the 
echo envelope to be described by 

V^(t) = V^(0)EXP[(-t/T2) + (-yVoT^/lZm^)], (5.8) 

where D is the atomic self-diffusion constant, G is the field gradient 
constant and m is the number of 180° pulses applied over the observed 
period. This procedure has the effect of refocusing the precessing 
spin ensemble at such a high rate that spins are kept in the observ- 
able ensemble even though self-diffusion through an inhomogeneous 
field is continuously, but slowly, changing the local fields of all 
spins. Diffusion damping of the envelope is made negligible by making 
m large enough. i 

The Carr-Purcell technique is particularly convenient because one 
may photograph the complete echo train displayed on the scope in a 
matter of seconds, whereas using the Hahn technique one must wait for 
the sample to re-establish thermal equilibrixim following each 90°-t-90° 
pulse sequence. One serious drawback is that the 180° pulse tuning is 
very critical because a small error is amplified m times over the 
course of the measurement. This leads to oscillatory echo envelopes 
which may lead to errors in T„. One should plot Ln[V (t)/V (O)] vs. t to 



135 

confirm a single exponential echo envelope. A modification of the 
Carr-Purcell technique by Meiboom and Gill (79) reduces the sensitivity 
to 180° pulse tuning by shifting the phase of the 180° pulses relative 
to that of the 90° pulse by tt/2. 

5.2.3. The Measurement of Linewidth and Second Moment 

Discussions of continuous wave measurement technique abound in the 
literature (see Abragam (7) or Andrew (8)), and we will not add to the 
surplus here, other than with a few brief comments of how the measure- 
ments were actually performed and data analyzed. Our equipment will 
be described in subsequent sections; suffice it to say that CW absorp- 
tion lines were obtained with a frequency-swept oscillator and magnetic 
field modulation was used. Typical lock- in detection was used giving 
one, as an output, the first derivative of the absorption line with 
respect to frequency. 

The linewidth, plotted in Fig. 18, is the half-width in frequency 
units between the points of Inflection (maxima and minima on the deri- 
vative curves) of the absorption curve. The second moment was cal- 
culated numerically, using the trapezoidal rule, directly from the 
derivative of the absorption curve. Each side of the line was cal- 
culated using the relationship, 

A. 



M^^ = 1 i 



3 



(5.9) 



Kf'((^.) 

1 ^ 



where F' (co) represents the derivative of the unnormalized absorption 

1 

curve, which is centered at w=0. Each error bar in Fig. 19 represents 
the difference 'in the values of M^ found for the respective sides of 

i 

the lines . 



136. 



All values of M„ calculated were corrected for magnetic modulation 
amplitude. The modulation frequencies used, 25 or 100 Hz, lead to a 
completely trivial frequency modulation correction to the second mo- 
ment. The correction formula presented in the equation below is a 
generalization by Halbach (80) of that due to Andrew (81) . We include 
the modulation frequency term, although it is negligible in our case. 



»,corrected ,,EXP 1 m 
^2 = ^2 - IT 



- f H^ (5.10) 

4 m 



The terms u and H in Eq. (5.10) are, respectively, the modulation 
frequency and one-half the peak-to-peak modulation amplitude. Modula- 
tion amplitudes were measured directly at the sample chamber. 

Great difficulty was experienced in obtaining a good signal in the 
region for 10 to 60 K. Presumably the increase in electrical conduc- 
tivity of copper in this temperature region leads to nearly complete 
exclusion of the alternating modulation field, even at a modulation 
frequency as low as 25 Hz . 

5.3. Electronic Apparatus 

5,3.1. Pulsed Spectrometer 

De Reggi (77) has presented quite a complete summary of what one 
desires of a pulse spectrometer. We will not repeat his summary; 
Instead, we launch directly into a description of the apparatus 

actually used for the experiment. Fig. 32 depicts in block form the 

I 
pulse apparatus. i 

We discussed the resonance phenomenon in Sec. 3.3.2., and pointed 

out that application of a 90° pulse required application of an intense 

rf voltage to the sample coil for a time to? which is inversely 



137 . 



proportional to the amplitude of the rf voltage. Continuous wave rf 
is supplied at a constant frequency and amplitude of about one volt 
peak-to-peak by the oscillator. The frequency of the rf should be 
approximately equal to that of the absorption line center frequency. 
The gating circuit, actually an integral part of the pulsed amplifier, 
will not connect the oscillator and power amplifier until a gating 
pulse is applied. While a gate pulse is applied, the power amplifier 
amplifies the one volt peak-to-peak input, and this rf pulse is applied 
to the sample coil via a "black box" coupling device. After a pulse 
the power amplifier is decoupled from the sample, and the nuclear sig- 
nal signal, induced in the sample coil, is amplified, detected and re- 
corded. The timing and pulse generating circuit provides the desired 
sequence of gating pulses, which are subsequently transformed into the 

desired sequence 90° rf pulses. 

Goted rf Power Amp. 



CVV rf OsciJlator 



r 1 



y . 



Timing B Pulse 
Generating Circuits 




Sample 
Coil 

Coupling 
Network 



:on^ 



Magnet- 
Sample Coil-'^^^^f ^ 



%. 



Triggered 
Recording 
Device 



t PA^e ct or _&_ Ppst_ Amp. 




jr. 




High Gain ( lOOdb) 
rf Voltage Amp. 



Fig. 32. BLOCK DIAGRAM OF THE PULSE SPECTROMETER ELECTRONICS 



138 



Continuous Wave Oscillator 

A General Radio Co. Type 805-D standard signal generator was used 
to supply carrier rf. The frequency output of this unit is continuous- 
ly variable from 16 kHz to 50 MHz. We measured the oscillator short 

/I 
term frequency stability to be +1 part in roughly 10 and the long 

term frequency stability to be roughly +50 parts in 10 over a 24 hour 

period. All frequency measurements x^ere made using a Hewlett-Packard 

5245L frequency counter. 

Gated Power Amplifier 

The gated amplifier used is a modified Arenberg Ultrasonic Labora- 
tor. Inc. Model PG-550-C pulsed oscillator. The unit was factory 
modified to perform as a gated amplifier rather than gated oscillator. 
Two additional stages of amplification were added, one gated, to in- 
crease carrier suppression and accommodate the rf input from the Gen- 
eral Radio type 805-D standard signal generator. This unit is capable 
of applying 300 V peak-to-peak pulses into a 50 ohm load. It is 
capable of pulses of maximum length roughly 200 ysec without appreci- 
able droop. Tx'7elve output coils provide a tuning range from 0.5 to 
60 MHz. We found that the coil marked for a specific frequency range 
did not always give one the maximum output in this range. 

De Reggi (77) mentions use of an additional gated low power ampli- 
fication stage to improve carrier suppression. It was our experience 
that this additional stage was unnecessary with the suppression now 
available through the use of low capacitance, high switching speed 
diodes designed for use in computer circuits. We use two crossed 
pairs of Falrchild 1N3600 diodes connected in series with the poxjer 
amplifier output. These diodes each have a capacitance of 2.5 pF; 



139 



Master 
Clock 




TEK 


62 








Waveform 
Generator 


















J 




A 





Recording 
Device 



/ 



TEK 163 

Pulse 

Generator 



TEK 163 

Pulse 

Generator 



Pulse 
Mixer S 
Selector 

Ref. (82) 



TEK 163 

Pulse 

Generator 



Pulse 
Mixer 6 
Selector 

Ref. (82) 



Wong 612 AT 

Pulse 

Generator 



Pulse 
Amp. 
Ref(77: 



To rf Power 
Amp. 
Fig. 33(g.). CONFIGURATION USED FOR 90'- T -90* PULSE SEQUENCE 



Baseline "*~ 



_ 1 a p. 1 J 4 1 » . > . s J *> p = [ > 

1 j . ^ .^ 

a 
\ 

a 
t 



t. T^ 



Fig. 33 (b.). TYPICAL DATA OUTPUT FROM 90-T-90° EXPERIMENT 



140. 



thus this arrangement presents a series impedance of roughly three 
thousand Ohms to a 20.8 MHz carrier, when they are not in a conducting 
state, and a very lox/ series impedance during application of a pulse. 
Pulse Program Circuitry 

Even for an experiment as uncomplicated as a 90°-t-90'' pulse 
sequence, one must contend with three distince time scales: 1.) The 
length of a 90° pulse, 2.) The time t between 90° pulses and 3.) The 
time interval between 90°-t-90° pulse sequences. The interval between 
90°-T-90° pulse sequences, during which the spin system re-establishes 
thermal equilibrium, should be no shorter than five times T . 

Our pulse program circuitry is for the most part of commercial 
design and construction. The units are designed to be highly versa- 
tile and provide one with the facility for a large array of pulse 
programs. In lieu of a list of the units, we illustrate the procedures 
used to obtain 90°-T-90° and Carr-Purcell pulse sequences. Fig. 33 (a.) 
illustrates the configuration used for generating a 90°-t-90° sequence. 
Obviously it is not unique, but was found to be a highly serviceable 
configuration. We found that generation of all pulses of a given 
width by a single pulse generator gave one a more stable sequence than 
using separately tuned units. 

The Wang 612AT programmed pulse generator is the heart of the 
system. This unit is capable of generating a sequence of from one to 
twelve equally spaced identical pulses plus from one to twelve equally 
spaced identical pulses, delayed from the first set. The width of 
the direct and delayed sets is independently variable over the contin- 
uous range from 1 to 10 usee, and the delay time is also continuously 
variable over this same range. This unit is particularly useful for 



141 

measuring T values of less than one second because one may increase 
T in a 90''-T-90° pulse sequence by eleven equally spaced Increments by 
simply sliding two switches. The master clock controls initiation of 
the complete 90''-t-90° pulse sequence; it is set to initiate a new 
- 90°-T-90° sequence each 5 T sec. One may trigger the recording device 

by any or all of the pulser imits. 

For T < 1 sec, we use a Tek 162 waveform generator as a clock 
source. It is set to repeat the 90''-t-90° sequence each 5 T sec. Only 
one of the Tek 163 pulse generators is triggered, and it in turn trig- 
gers the Wang 612AT which generates the 90°-t-90° sequence. A Tek- 
tronix type 585A oscilloscope is triggered by each 90° pulse from the 
612AT, and a segment of the FID after each of the pulses is photograph- 
ed. The segment photographed is an intensified portion of the trace, 
and it always has the same temporal relationship with the leading edge 
of a 90° pulse. For each value of t, a minimum of three 90°-t-90° 
pulse sequences was photographed, and the number of signal averaging 
from this procedure. Fig. 33(b.) illustrates the typical data output. 

For 1 sec < T^ < 10 sec, two Tek 163 pulse generators were used to 
generate the 90°-t-90° timing sequence. One unit was triggered at 
the leading edge and the other, near the end of a sawtooth oscillation, 
provided by a Tek 162, The pulse from each Tek 163 then triggers a 
single pulse from the Wang 612AT. The time t between 90° pulses in 
J the sequence is determined by the length of the sawtooth waveform from 

the Tek 162, w^hich is triggered manually. 

Fig. 34 (a.) illustrates the configuration used for a Carr-Purcell 
sequence. The actual pulse sequence obtained is illustrated in 
Fig. 34 (b.), and 34 (c.) depicts a typical echo envelope as photographed 



142 



; 



Master 
Clock 




TEK 163 


® 


Pulse 
Mixer a 
Selector 




i 


Recording 










Device 


















Pulse 
Mixer a 
Selector 






@ 
















1 




Delayed Gate 


(D 


TEK 162 


D 


TEK 163 




Generator 











ng.34(a). CONFIGURATION USED FOR CARR-PURCELL SEQUENCE 



® 



5)' 



>5T. 



y 



/ 



/ 



Gotes TEK 162 



Alt i.^o° P>*!«s 



@ 



Fig. 34 (b.) OUTPUT AT POINTS INDICATED IN Fig. 34 (o.) 



^1 


\l\h, 


















M 


'^^^ 




fi^iihi 




- 
















j.j 


m\j\ 


%^ 


^ 


\AA,'\A 






















l'v\/v 


w\^ 



T = 2msec 



Fig.34(c.) TYPICAL DATA OUTPUT FROM . CARR- PURCELL SEQUENCE 



1A3 



from the oscilloscope. The Princeton Applied Research, Inc. waveform 
eductor was used primarily as a delayed gate to gate the Tek 162 on 
for a predetermined time, during which a number of 180° pulses from a 
Tek 163 Xv^ere generated. The number of 180° pulses generated during 
the gate pulse is governed by the sawtooth repetition rate set on the 
Tek 162. Figs. 34 (a.) and (b . ) are self-explanatory so we will not 
belabor the reader with additional words of explanation. 
Data Recording Device 

All data were recorded on type 47 polaroid film from the screen of 
a Tektronix type 585A oscilloscope using a type C-12 Textronix trace 
recording camera. A tektronix type 86 plug-in preamplifier accepted 
the output of our receiver. 
Magnet 

The magnet used was a 12 inch Varian type V-4012-3B, fitted with 
factory supplied cylindrical pole caps which provide a 3.5 inch air 
gap. With this pole cap configuration, the maximum obtainable field 
is approximately 9 kG; and the factory specified minimimi field homo- 
geneity is roughly 50 mG/cm over a circular area 2 cm in diameter. 
Frequency and Time Measurements 

All frequency and time measurements were made using a Hewlett- 
Packard model 5245L counter equipped with a type 5262A time interval 
unit. These units allowed very precise measurement of both the time 
interval between pulses and pulse width. 
The Coupling Network and Receiver 

The coupling network and receiver are considered together because 
they are so intimately connected. More properly, perhaps, one should 
divide this discussion into three distinct sections. Taken in order 



144 



from the sample coil these are: a.) The coupling network and rf pre- 
amplifier, b.) the wideband rf amplifier and c.) the detector cir- 
cuitry and postaraplifier. The coupling network and preamplifier are 
burdened with a most difficult task: That of coupling a '^ 300 V rf 
pulse from the power amplifier to the sample coil during a pulse, then 
recovering rapidly from the pulse to couple a high gain amplifier to 
the sample coil in short enough time to observe a FID, which may be 
less than 20 ysec long and have an amplitude at the coil of only a few 
tens of a microvolt. The wideband amplifier must provide sufficient 
voltage gain, as supplement to that of the preamplifier, to render the 
nuclear signal visible. The detector removes the information carried 
by the amplitude modulated rf carrier, and the postaraplifier primarily 
isolates the detector circuit from recording devices. 

The requirements placed on the detector and postampllfier are 
quite stringent. The detector must accept the amplitude modulated 
carrier, modulated by the lineshape function expressed in the time 
domain, and remove the information in such a manner as to introduce 
no distortion. The detection process is basically a rectifying and 
filtering process whereby, ideally, the detector is perfectly linear 
and the filter passes only frequency components contained in the FID. 
Normal diode detection is only suitable if the diodes are forward 
biased into a region of high linearity. This is done most effectively 
through the use of phase coherent detection [see the following re- 
ferences, taken at random from the literature: (83), (84), (85), and 
(86)], which has the added advantage of an intrinsic increase in the 
observed signal to noise ratio. Ref. (86) presents a very good 
discussion of phase coherent detection. Unfortunately, our General 



145 



Radio Co. type 805-D signal generator lacks the short term stability 
necessary for phase coherent detection; therefore, we were constricted 
to use of diode rectification. Distortion will, of course, be intro- 
duced if the detector and postamplifier have too narrow a bandwidth 
to pass all the Fourier frequency components present in the FID. 

The requirements placed on the wideband amplifier are similar to 
those above. It must linearly amplify the rf and have a bandwidth, 
about the central Larmor frequency, wide enough to accept all frequency 
components present in the FID. Ideally, the bandwidth of this ampli- 
fier is such that it will provide roughly 80 db gain over the total 
range of frequency at which one may wish to work. 

For our work in liquid NH„, we used a Lei, Inc. model 5182 ampli- 
fier, factory modified to provide roughly 120 db overall gain at a 
center frequency of 3 MHz. The bandvridth of this unit is '~ 1 MHz. 
Modifications of the input stage will be described subsequently. We 
utilized an Arenberg Ultrasonic Laboratory, Inc. WA-500E wideband 
amplifier and PA-620-L narrow band preamplifier for our work in solid 
NH_. -The rf gain of the WA-600E unit is 60-65 db, flat to within 
+3 db from 2 to 65 MHz. The detector, postamplifier circuit has a 
3 MHz bandwidth and contributes an additional 20 to 25 db gain. 

We now consider the coupling network and preamplifier. The prob- 
lems involved in coupling the sample coil to power amplifier and 
receiver are manifold. There are the follov/ing three relatively in- 
dependent states of the system that one must contend with: 1.) The 
application of an rf pulse, 2.) the time t , following the applica- 
tion of a pulse, during which the electronic system recovers from the 
pulse and 3.) the period of observation of the FID. 



) 



146 



In the ideal case one would have the rf pulse V (t) conform to an 
equation of the form, 

sin(a) t + 4>) It I < — 

° (5.11) 



V (t) = 
P 



I Nit 
o 



where w is the Larmor frequency corresponding to the center of the 

absorption line and N is the ntimber of complete cycles in the pulse. 

Let (j)=0 for sake of simplicity. The ubiquitous Heisenberg uncertainty 

principle, when applied to this situation, predicts that the uncertainty 

in frequency w of our pulse is roughly the inverse of the length of the 

pulse. Cast in perhaps more concrete terminology, that of a frequency 

spectrum of V (t) , the uncertainty in frequency becomes more apparent. 

We appeal to the technique of Fourier transform analysis to transform 

V (t) to the frequency domain, and find (87) 

sin(aj - u) (Ntt/o) ) 

o 

for large o) and cj close to to . The constant a' is a product of the 
^ o o 

Fourier transform and not important. The function g(a)) also has a 
width of roughly the inverse of the length, in time, of the pulse. 
Thus, we see that in order to excite all nuclei in a solid, with a 
characteristic NMR distribution function f(a)) of width 10 Hz, our 
pulse width should be less than 10 psec; and, furthermore, the pulse 
should be presented relatively undistorted across the sample coil. 
This condition requires that the bandwidth of the circuit seen by the 
power amplifier, the circuit containing the sample coil, be greater 
than 0.1 MHz. In addition, for maximum power transfer one must try to 



; 



147 



match load impedance to that of the povrer amplifier. 

One may place the sample coil in a resonant paralled tuned circuit 
as shown in Fig. 35. The power amplifier is represented by a loss-less 
voltage generator in series with a characteristic output impedance Z . 
The sample coil is represented as a pure inductive reactance L in series 
with its intrinsic resistive component R . The 3 db bandwidth of a 
parallel tuned circuit at resonance is represented by 

B = 0) /2ttQo, (5.13) 

where the coil quality factor is defined by, 

u) L 
Q»=-|-^. (5.14) 

s 

The conditions on the bandwidth force certain limits on the quality 
factor Qo of the sample coil. For example at oj = 27r x 20 x 10 , a 
pulsewidth of 1 usee requires that Qo be less than 20. One must of 
course also make certain that the receiver input impedance Z is much 
greater than that of the sample coil network. 

In the second time period, the energy stored in the L C network 
must be dissipated, primarily in R . This ringing of the tank cir- 
cuit decays, in a resonant parallel tuned circuit, with a time constant 

t = 2Q/a) = 1/ttB, i.e. , 
r o 

V (t) = V(0)EXP[-TrBt]. (5.15) 

r 

If V(0) = 100 V and ttB = 10 , it takes the ringing voltage 16 usee to 
decay to the '^^ 10 yV amplitude of the signal. Clearly it is advanta- 
geous during the ringing period to have as small a sample coil Qo as 
possible. Damping of the sample coil ringing is sometimes accomplished 
via an active device, but we simply relied on decreasing the coil Q», 
via an adjustable resistance paralleled with the sample coil, to the 



148 



) 




-^^ Crossed Diode Pair 



Sample 
Coil , 



' i »■ 



rf 

Power Amp. 



-^c 




Receiver 



.Fig. 35, TYPICAL PARALLEL TUNED SAMPLE COIL CIRCUIT 



To rf Power Amp. 



R. 



#^ 




js Transformer 



n 



Fig. 36. LEL INPUT CIRCUIT 



To rf Power Amp. 




6922 



^Rt 



Fig. 37. ARENBURG INPUT CIRCUIT 



J 



149 



point where the FID could be observed following decay of the ringing. 
V/hen the voltage across the diode pairs drops below '^0.5 V, they 
switch into the high impedance state; thereby effectively decoupling 
the tank circuit from the power amplifier. One then has the situation 
illustrated in Fig. 35, but with the diodes acting like an open switch. 
We tune C to resonate with L ,and all stray reactance, with the cir- 
' cuit in this state. This tuning is done by applying a very small 

amplitude modulated rf signal at to through the diodes, which remain in 
a high Impedance state. The capacitor C and receiver are tuned to 
maximize the signal. The parallel tuned resonant circuit is particular- 
ly desireable for detecting the nuclear signal V (t) because the vol- 

n 

tage developed across the capacitor C is not V (t), but Q times 

n o 

V (t) . Clearly then, one would hope for the largest possible Q , 
consistent with FID bandwidth requirements, during detection of the 
FID. One additional factor should also be taken into consideration — 
for each input tube, there exists an optimum input source impedance at 
each operating frequency. The reader who is interested in optimizing 
the signal-to-noise ratio (S/N) should consult references (88), (89), 
and (90). 

Follox^ing S/N optimization techniques in re-designing the Lei in- 
put circuit, increased the S/N by 7 db over the circuit shown in 
ref. (77). The coupling circuit used with the Lei model 5182 is 
^ " shown in Fig. 36. Our design incorporates the sample coil into the 

actual input circuit, thereby eliminating additional circuit elements 
which compromise performance for the sake of -impedance matching to 
a 50 fi load. This circuit provides sensitivity enough to easily 
observe the pure quadrupole FID in a sample 3 cm long by 1 cm in 



; 



150 



diameter of Hexamethylene tetramine at room temperature. The importance 
of source impedance optimization cannot be overstressed when one must 
contend with very weak signals. Incidentally, a recovery time of 
'■^ 30 ysec was obtained with this circuit at 3.3 MHz; additional diode 
pairs did not make any noticeable improvement in the recovery time. 

A similar circuit was used in conjunction with the Arenburg model 
PA-620-L preamplifier (see Fig. 37). The relatively strong proton re- 
sonance did net require much attention to matching source impedance for 
optimum S/N. We found the S/N at 21 MHz to be much better than that 
at other frequencies. Ringing vtas quite a problem due to the relative- 
ly short proton T^ of 10 ysec in solid NH~, and required severe damp- 
ing of the sample coil. Possibly one of the active coil damping cir- 
cuits in the literature would have helped in this instance. Total 
dead time from the end of the pulse was roughly 5 ysec. 

This brief discussion does not scratch the surface of a very in- 
teresting and challenging electronics problem. The reader who wishes 
to investigate this problem in more detail is referred to the litera- 
ture; in particular, references (91), (92), (93), (94), (95), (96), 
(97) and (98) may be of help. 

5.3.2. The Continuous Wave Spectrometer 

The continuous wave spectrometer was used in the determination of 
linewidth and second moment. The spectrometer used is of rather con- 
/ ventional design and is illustrated in the operational configuration 

in Fig. 38. We used the frequency swept, magnetic modulation mode 
for all absorption line measurements. 

The heart of the system is a slightly modified Robinson type 
oscillator (99) constructed in this laboratory by P. Canepa (100) . 



151 



J 



Voltage 

Ramp 

Generator 



Magnet 



Sample Coil 




Modulations""^ 
Coils 




Frequency 
Counter 



Robinson Oscillator 



i. 



Audio Power Amp. 







PMMt 

OeTtC-Tetv 



^ "" P " " I * ft 












QfrMO RC FluTl(\' 



Lock-In Amplifier 



Audio Oscillator 



Chart Recorder 



Fig. 38. BLOCK DIAGRAM OF THE CONTINUOUS WAVE APPARATUS 



152 



This vinit is capable of the very low level rf (<1 to 100 mV) necessary 
for work with samples having long T . The frequency sweep is effected 
through application of slowly varying ramp voltage to a voltage vari- 
able capacitor. The ramp was generated through use of a Tektronix type 
operational amplifier, operated in the typical integration configura- 
tion. An Electronics, Missiles, and Communications, Inc. model EJB 
lock-in amplifier accepted the detected rf (audio) output from the 
Robinson oscillator, and the processes absorption signal was recorded 
on a Brown chart recorder. A Hewlett-Packard model 201-C audio oscil- 
lator supplied audio to both the lock-in reference channel and a Ling 
Electronics model TP-100-2 power amplifier. The Ling model TP-100-2 
drove a Helmholtz pair of coils wrapped directly around the magnet 
pole faces. The maximum field modulation amplitude possible in the 
air gap of the magnet was roughly 16 G at 25 Hz and 10 G at 100 Hz. 
A Hewlett-Packard model 524 C/D electronic counter, equipped with an 
automatic frequency marker, was used to measure rf frequency. 



-..i^AWrtr— ^r«c-rf^.«— Vt*.»*^ -•-■ ■*_.■■'..- liJ A_>t >•■»■■ *.r^''*'»«. *,* ■ -H.* --"-rtW-aMfc^K.-.-,! 



) 



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BIOGRAPHICAL SKETCH 

James Lynn Carolan was born August 15, 1943, at Long Beach, Cali- 
fornia. He attended Beaumont High School in Beaumont, Texas and grad- 
uated in May, 1961. In September, 1961, he enrolled in Lamar State 
College of Technology in Beaumont, Texas and received the degree of 
Bachelor of Science with a major in physics in May, 1965. He enrolled 
in the Graduate School of the University of Florida in September, 1965, 
at which time he was granted a graduate assistantship in the Department 
of Physics. Laboratory teaching duties and course work occupied his 
time until December, 1966, at which time he elected to continue work 
directed toward the degree of Doctor of Philosophy, by-passing the 
degree of Master of Science. His work from December, 1966 until the ' 
present consisted solely of satisfying the requirements for the degree 
of Doctor of Philosophy. 

James L^Tin Carolan was married to the former Joan Marion Satzer in 
December, 1965, and they expect their first child in September, 1969. 
He is a member of Sigma Pi Sigma and a junior member of the American 
Association of Physics Teachers . 



158 



) 



This dissertation was prepared under the direction of the chairman 
of the candidate's supervisory committee and has been approved by all 
the members of that corarnittee. It was submitted to the Dean of the 
College of Arts and Sciences and to the Graduate Council, and was ap- 
proved as partial fulfillment of the requirements for the degree of 
Doctor of Philosophy. 



August, 1969 






^ M 



Dean, College/ 6f, Arts and Sciences 



-y 



1^ 



C?vH^ 



Chairman: T. A. Scott 
A. A. Broyle's 



^■M-¥^{/fr/. 



E. H. Hadlock 



Dean, Graduate School 




jl. S. Rosenshein